John,
You may have to use some kind of lubricant for the sliding top. Depends how bad it is sticking, how much "stick" you are willing to live with, how much you want to avoid any kind of lubricant. Don't use any petroleum or WD40 in any case! Meguiar's #26 Yellow Wax works great on interior wood. If the wood is peeling you may be a bit beyond the help of just wax. You might try applying some adhesive under the peeling edges but be VERY careful not to get adhesive everywhere. If the wood is peeling at the edges (by the rail slats) then that may be what is causing it to be sticky. If you can fit a q-tip or something similarly small into the groove the cover slides in, then agitate to try and remove any foreign debris. (move panel as far back as possible, clean, move back; repeat as necessary) That might help the "stickiness". Good luck, let us know how it turns out.

Lee