@Captain Hindsight - Disregard what's written above.
First of all - dads taking paternal leave is very common. This isn't 1970.
Secondly, all FMLA does is secure your job when out on medical leave. It has nothing to do with pay. The only way you'll owe the company money if you leave is if you're being paid with PTO that you haven't yet accrued. For example, where I work, hourly employees get 24 PTO days per year ... so they accrue two days/month. If someone wants to take 10 PTO days in January, they can - even though they're only accruing two days at the end of January. If they quit on Feb. 1st, then 8 days (64 hours) of PTO would come out of their last paycheck because they took those days off using PTO and were paid for them before they'd accrued the time. Either way, it has absolutely nothing to do with FMLA.
Lastly, if you're honest with your future employer about your paternal leave, they will understand and - if they're a decent company - encourage you to take what you need prior to starting. That said, I'll refer to the previous paragraph .... if you're using your PTO to get paid during leave, then you may want to consider cutting your leave short and starting your new job earlier, strictly for financial reasons.
Best of luck to you and congrats on the baby!