Congrats. Which country/state? There are laws/regulations related to leaves. FMLA allows for 12 weeks if I recall correctly. I would suggest talking to HR and specifically ask about paternity leave. Looking elsewhere may not lead to a better outcome unless you can close within the next two months and defer start date to/until after new arrival. If you like where you are then looking elsewhere may not be a benefit.
As a father, I would suggest spreading out the leave esp. if everything goes normally. The first couple of weeks for sure but afterwards break it out to give some time for everyone to adjust.