First thing, this shall pass. Your son will get older and more independent soon enough. Yes, it's hard right now, but it will get better. My oldest literally only slept 40 minutes at a time for the first nine months of his life. It was so hard. But now? He sleeps 8:30 to 6:30 with only one wake-up at night. He is so much better and yours will be as well. I know it's hard, but you can do this.
Secondly, make sure your birth control is locked down. Get an IUD. You do not want to have another child with your husband. If he's already not don't anything, it won't get better.
Thirdly, I agree with making an exit plan. You have a job, that's great!! You're financially independent of your husband. Start looking for a better job. You don't need to leave your husband today, but you can start making a plan to leave.
First, look up how much child support you would get from your husband. There are online calculators.
Could you afford a studio apartment in your city? Your son is young and doesn't need a lot of room yet. You can live in a studio apartment while he's young to save money and move when he's older.
Daycare costs less as your child gets older. Once he is one, it becomes a little more manageable. Look into home daycares. They are cheaper. Get recommendations on your city's reddit page.
Also, your life would be easier with one less person messing up the apartment, creating dishes and laundry. And your could be more mentally healthy due to not hoping he will change.
I'm so sorry you're going through this. I'm sorry your husband is failing to be a good partner or parent. Hugs.