I had to re do the bathroom door in my in laws moho. I was able to carefuly slice the skin from the door. then I re-glued it back on the frame after I sanded it.
On exterior doors(inside surface) you can buy a luan panel at the local HD or buy an oak panel at a local lumber yard, make sure the center is masonite(MSD) not plywood. If you take one of your drawers down to the local paint store they can match the color of the stain or as I like to do paint.
I use primer. Be sure to thin both the primer and paint. You will need a flat surface to work on with a blanket to protect the exterior paint. Just take your time and it will look just fine. What I like to do is 1 thin coat of primer, sand and then paint 3 thin coats of flat paint and then use a 3 coats of waterbase clear coat over it. Be sure to tackle rag between coats.
May your black water hose never break! Life Charter member since 89