Jerry, Folks have been dealing with veneer overlays for years and years. If you are a purest, then follow the procedures listed in our owner links & threads. The issue IMHO is the solid wood center panels and the veneer overlays have 2 different expansion rates and the weaker (veneer) is inclined to curl on the edges.
I happen to LOVE cherry wood because of the way it "tans" and ages and reveals the character of the wood. I removed the veneer overlays with a little lacquer thinner and peeled it off. Sand with 220 and stain with Zar cherry and then apply the glaze finish. I am still experimenting with various methods and will advise if I ever succeed.
Blessings, Joy and Adventures in the coming year! Fred