@cabinetman... thank you for tip of using the peel and stick wood banding. I think this will finish them off nicely and then they will be ready for painting. At first I couldn't find any that I felt were wide enough since they come pretty standard @ 2" and lower. I finally found a 5" wide unfinished maple.... PERFECT.
Depending on where you buy it, a usual width is 24" by 8' long. That would be the least expensive way to purchase. As for installing, cut strips to allow an overhang of a minimum of ľ". You will need the overhang to allow for placement. With PSA (pressure sensitive adhesive) wood tape, it's like using contact cement, without the smell of glue.
For applying the strips, the outer stiles just lay on top to bottom. For the rail pieces, once you have sized them for width, have a helper hold one edge to the edge of one of the stiles, and mark the other end where it meets the edge of the other stile. You can cut a straight line using a combo square and a sharp utility knife. You may allow a smidgen on the length. Test it before peeling it. You don't want to lay a strip right on the old seam of where a rail meets a stile. In your case by adding strips, your seam will be beyond that. With your experience I wouldn't try cutting the veneer to keep the joint back of the amount of the new add on area.
For trimming the edge, use a plain mill file, like the one to file plastic laminate. You will notice that the edges of the file are milled like a rasp. Lay the file on the edge perpendicular to the face. IOW, when filing off the overhang on a face frame, start with the file on the edge, with the far end of the file tipped slightly up.
Use the edge of the file in short sharp strokes only on the down stroke, to rasp off the veneer. You want the handle of the file held back in the direction that you are filing. IOW, a slight angle of the file to the veneer will cut the edge smoother than having the file straight up and down. Once you try this out, you'll find it's really the fastest way to trim the veneer. If you use a utility knife or a flush trim router bit, it can catch the grain and cause a tearout. Once most of the bulk of the overhang is off, you can go back and dress the edge with the face of the file, at about a 10 degree angle or less. You will get the hang of it pretty quick.