Hi - four weeks is abysmal. Sorry to hear that. I am not a lawyer but have had three kids in DC.
FMLA protects your job. It requires employers to hold your job, but not provide any kind of leave.
DC paid leave offers you some pay, up to about 1000 dollars a week depending on your salary but this is the max, for up to 12 weeks. Some employers supplement this leave to provide additional pay but sounds like yours will not.
You should clarify that you will be able to stack your leave - that is, that you can use 4 weeks of the company leave, then 12 weeks of DC leave.
You mentioned STD. This can be used immediately following the birth, generally for 4-6 weeks depending on mode of delivery. Often this is partially paid leave (at my company it’s 66 percent of your salary). You also need to clarify if you can stack this leave because this will go beyond the amount of time that FMLA holds your job, but many employers allow this.
Best case scenario it sounds like you could be able to stack STD (4-6 weeks), DC paid leave (12 weeks at up to 1000 a week) and your company parental leave (4 weeks). Worst case these would all run concurrently and you’d be capped out whenever FMLA ends. Federal is 12 weeks, DC may be 16, I can’t remember.
You also need to check if there’s a waiting period required by the potential new employer to access benefits. Sometimes they are only offered to employees after one year etc.
Hope this helps