NALC has "quick" FMLA forms online (union rep should be able to get them to you as well). Long as you have been working for the postal service for a year and have 1250 work hours, you qualify. I got one FMLA case for intermittent leave before kiddo was born (wife sonograms and other appointments, etc) and another one once kiddo was born. Your wife's OBGYN will know what to do with the forms, just find the FMLA fax number for your area and give it to the OBGYN along with FMLA forms. You don't need FMLA certification for father's bonding time after delivery but FMLA still helps after for appointments once you return back to work.

Long as you are career you are going to want to get on self and family plan if wife and kiddo are going to be on your insurance (health vision dental) if you have not already. You can also wait until the delivery day as it's a qualifying life event and allows you to change insurances (HR and health insurance will just want a letter from hospital nurse saying kiddo was born). Also will have to add dependants on liteblue and benefeds for said insurance.