Stay at your job. Milk them for whatever paternity leave they will give you. And if you announce you are going on a long paternity leave, they are not going to fire you right before you leave. DO NOT SWITCH JOBS NOW.
When the hospital sends you the bill - argue, argue, argue. They did stupid stuff to us like bill both my wife and the child for all the same delivery line items ("guys, it was the same procedure!"). Get it to a reasonable number and put it on a payment plan.
Use the paternity leave to start looking for jobs. You'll have time and a clear head. And a much bigger perspective on life. You may even find that with the change, you were missed at work and that the burnout seems far and distant.
After that, having a kid is cheap. (At least at first). Get yourself a Costco membership. The first year we spent maybe $35 a month on diapers and $45 a month on formula. Clothes and furniture and toys other people will be stoked to provide for you. At tax season, enjoy getting a few thousand extra dollars back.
Welcome to the next phase of your life. I wish you and your growing family much happiness.