Graduating/job searching while expecting?

Hi there! Looking for folks with insight on changing jobs with a baby on the way. My wife and I are just about to do our first embryo transfer (she is carrying if it matters). If this or the next one stick, that will put us at a due date in mid-Fall. Yay! The thing is, I'll graduate from my masters program in early May and my on-campus job will end. I'm obviously starting to look/apply at jobs (non-profit, hopefully on the higher paying end) but feeling very nervous about how to handle the timing and negotiation for the kind of leave I want. To have the start to parenting that I want, I feel very strongly about taking at least 2-3 months leave when baby is born, not staggering my wife's and my leaves, though I know some people prefer this. I'd feel very sad to miss out on those early days, regardless of who is carrying the baby.

If the timing works as we expect, I'll be in new a job under a year before baby is born, so not legally guaranteed FMLA. We have a little, but not a ton of financial flexibility. I think we could make it work if I was completely out of work after baby is born, but I believe my wife only gets a % of her salary, so it's not ideal. I'm open to all sorts of options: short term work before baby is born, full job search after, etc. My ideal situation is to have a job that gives me paid leave when I want it (lol) and then go back to that same job after, but that seems like it might be hard to find. To complicate matters, I'd also love to stay at home part time with baby, but securing some kind of income comes first. If necessary, I'm also willing to take a job knowing I will have to quit if they can't accommodate what I need. I do have folks who will be able to help me network and strategize, but I'm not yet ready to clue them in on the specific constraints of my situation.

Does anyone have any insight on this kind of situation, especially, but not exclusively, for NGPs? Specifically, I wonder about finding accommodating employers (may be industry specific), when/how to talk to prospective employers about my situation, how to protect myself from discrimination, or any arrangements I might not have thought of. Respectfully, please no suggestions to shift our timeline or not take the kind of leave that I need. Thanks in advance!