You might qualify for aid once the baby is born, since you’ll be going from a two-person family to three.
I think it’s definitely worth looking into privately-rented places. We tend to avoid them because we like maintenance to be readily available, but if you can find a good landlord, I’ve come across a lot who don’t seem to mind cats (we’ve moved a lot).
Personally, I would figure out whatever is going to cost less and do that. We ended up having to move too, and then I lost my job while on maternity leave, and we’ve had tons of unanticipated baby-related expenses, even though we saved and planned. It happens. Do a lot of research, see if you can find someplace further away that is cheaper with utilities included, account for increased transportation expenses, do the math, and see if it’s worth it.
How much longer until your husband graduates? Is his promotion just hoped for or expected? Do you plan on working or staying home with the baby? Could he get a second job when he is done? $350 a month increase is a lot, on top of health insurance increases, diapers, etc. One thing I’d consider is your laundry situation. If you have it included somehow now, I’d strongly consider keeping it. Out hot water and electricity use has increased a ton over the past year.