But probably for different reasons.

I think you should get a job as quickly as possible so you are able to self support. Eventually you will snap and it may be the end of the relationship. Then no choice for his kid... bus and aftercare it is! And since you will be picking up baby, husband can handle HIS kid's pickup or find someone else to do it. Also, since your time is now limited, husband AND his kid can BOTH start helping around the house. Be sure to allow for plenty to go into your (hidden) savings... you likely will be needing it.

I'm assuming you don't want to just leave so here's an answer based on if you want to make this work-

Therapy- first and foremost for you. Both to get support but also family therapist can offer a lot help in how to deal with difficult children. My ss had a druggie felon mom. I would really encourage the child to be in therapy as well, he's only 9 and definitely got trauma that's likely coming out as bad behavior, but there's still a chance to help him at that age. parenting a traumatized emotionally damaged kid isn't going to be like dealing with a regular child, you'll need different strategies and guidance. With the help of therapists I would say it took a bit over a year to work through my ss worst behavior problems after his mom got arrested and he came to live with us. He was 12. He still has a lot trauma that's causing him to act out now as a young adult but for a time he got much better with regular weekly therapy sessions. for the years he lived with us he did get a lot better, good grades, good friends, helpful and our life was more peaceful, until his mom came back but that's another story. Therapy with someone trained in both family and trauma. 
Your dh needs to recruit you some help- can his parents lend a hand? He also is going to need to learn that babying his son doesn't make up for what the mother did. It just reinforces that acting out is ok. The kid probably is in a lot of emotional turmoil from all these changes and his mother abandoning him, he needs his dad to show him it's going to be okay. In reality, your husband should change to a career that will let him be home. I don't know how he thinks that he can keep his family together being away months at a time. I wouldn't be surprised if the boy acts out when his dad leaves because he feels abandoned all over again

And the deployment work ups & deployment itself are hard on military families.  Other posters have given you some good advise.  I will add are you near a military facility?  If they have daycare get on the wait list.  They often have drop in slots as well.  Use them to give yourself a much needed break.  Second if your husband is deployed there may be additional special pays during the deployment period....depending on the service/deployment location his pay may also be tax free.  Find out.  Maybe there is no need for you to go back to work immediately.  Does his squadron or flight have a family support group?  If so you may want to consider reaching out so that you can build a local support network.  When my kids were young that group acted as a childcare collective where we would support each other with childcare when needed.  It can also provide a social outlet.

Your SS needs counseling & Tricare will pay for it.  Insist that your DH get the ball rolling on that.  Your SS should go to his grandparents or other family members while your DH is deployed but if that can't happen you need to make it clear that you are going to parent as you see fit.  Take steps to protect yourself.  Install cameras in common areas, put sensors on the kids bedroom doors so you know if SS is roaming at night.  Can your baby/babies sleep with you?  If so I would do that as well.

and your DH is no Officer & Gentleman if he will not take steps to help his son who is obviously hurting and protect his pregnant wife and children.

Try to find some place that offers help you need.. This is not a life anyone wants to live.  Why don't SS Have activities to get him out of tge house?   You have a choice to make.  
'SS  isn't going away, DH has no place to dump him.  You are the only one who can move on.  You love DH, buy can't live in thus changing relationship.  It's unfair for DH to expect you to take care of his son,  expecially a son who needs a lot of help.  Why isn't he seeing someone.?

Definintely take the therapy route.

If you feel you are snapping you need some kind of support system - now that you are in a more stable location, start building your mommy network as others have suggested.

Now is the time to be strong for your little one.

I feel kind of bad for SS. But if you truly feel that he is just born bad and there is no fixing, consider getting cameras for the household.

a paternity test done. If SS is not his, that gives you some potential leverage.

Once DH is deployed, call SS's Maternal or even Paternal GPs and drop him off for the holidays.  

I too agree that you should go to work.  To be independent of DH's financial control.

Build your new life structure while DH is deployed and when he gets back, roll out the paternity results... if DH is not SS's BioDad, and define the boundaries that DH will stay in when he gets back.

I was never in the military. Though I have worked a number of extended assignments domestically and internationally while DW was home with SS and my SIL when DW had guardianship of SIL.  I would fly in for weekends and it was an adjustment for all of us. The three of them had their routine, I was not a part of it. So, I start being DH and dad, and the decisions I was making were not how DW was leading the home in my absence.  So, I adjusted.

When DH gets back, inform him on the new normal.

If SS is in fact DH's, don't mention the paternity test.  Just use it to give you clarity and guide your vectors on this marriage and family.  Anyway SS's paternity works out, you set the standards of behavior and performance for SS and he complies. Tolerate nothing but compliance from him. If he fights it, make his life one of escalating abject misery.  Take electronics, stand him in a corner with his nose planted firmly, he gets shitty, he suffers.  Do not over think it, just enforce the standards.  DH gets no say. After all, he won't be there.

Take care of you, take care of your LO.

That's not a great position to be in. Wow, you are in a really tough spot.

I think you need to have a heart to heart with your husband and let him know some home truths and that if he continues to put all the child care responsibilities on your shoulders without any authority or outside support you will resent him and his son and that could be disastorous for your relationship and for the wellbeing of his son. If he wants to come home to a wife and a son who is thriving then he needs to step up and help find solutions that will make things easier for you.

Okay I feel your pain!  You need to use every available military resource that you can find.  Therapy for you, and the family.  Childcare - you need a support system.  Other military wives, groups.  Your hubby needs a wake-up call plus a boot up his bum so his head can get out of there.  There need to be ground rules as a family and everyone has to help.  You need to work to be financially independent - set up an account for yourself in the bank - women need a nest egg as changes can occur.  Stepkids can tear you and your marriage apart as they can be so manipulative - you have to be more cunning, smarter and thwart this behavior!  So the house isn't clean, the laundry isn't done, take-out food -- those boys have to be more independent.  

Step up a master calendar and a chore list for everyone.  You doing everything with a newborn is ridiculous.  Make a plan and everybody including your SS has to adhere.  It not, be prepared to say this no longer works for me and run, don't walk.  Thus the reason to have a job, a support system, and good therapy.  Don't sell yourself short and work yourself to death.   Take care and sending you a big hug!

I am onboard with the therapy idea. Soon. Fast. You have way too much on your plate and what you say about your SS scares me (in that he scares me). 

It does some like your DH is also dealing with a ton on his plate at the moment as well, but that does not make it fair for him to expect you to raise his son for him.  Which is then even that much more difficult with the added stress of having to do this potentially while he is deployed, so it's truly all you and he isn't even around...

I am also sorry to hear that you are not able to see your daughter very often.  That sounds very hard to accept long term, and I understand how you feel when something distracts from the opportunities you do have to spend time with her.  Your daughter is very young, and you have a chance to make decisions and changes in what you prioritize in your own life to be more a part of her life.

I honestly think it would be heartwrenching to raise a kid who you feel no attachement to full time, who is the same age as your daughter, while you only get to see her once a month.  I don't think that sounds tenable as a way to structure your life to me, and I would consider taking action to work towards changing how that is structured.