Do I sand the final coat of paint before using poly on cabinet doors?

You should sand between every coat of paint and the poly. Just wipe it down good too

I don't know how many coats of paint you sprayed on! It's possible you are spraying too thick coats. Several light coats will get you a smooth finish. Three plus will get a smother finish. No sanding required. I personally prefer using a small foam roller and foam brush' Smooth finish, no brush marks. Do not sand the paint before applying poly. At least three coats of poly. Lightly sand in between poly coats. I like to use a green Scoth Brite pad.

Sand the second coat.

Sanding is correct but you must use a fine sandpaper like 220 grit on the first coat, 320 or so on the second and then 400 or up after the last. Use must use a tacky cloth or microbiber between layers. Don't shake your paint and you don't have to stir your paint very much if at all. The air bubbles will cause a rough finish. Make sure the area is dust free and there is no breeze to kick some up. Last and very important make sure your brush is clean and dry. To much cleaning product left in the brush can disrupt the finish.

Thank you I may not of had the adjuster on the spray gun correctly lol so back at it tomorrow hopefully better results this time. So far only one coat on five doors backside so havent caused to much damage yet fingers crossed for the rest.