Depending on the size, layout, and elegance
of your shop cabinets, you might try sliding
doors.
These, of course, can be as fancy you like.
If large, rollers in a track are helpful like pocket
or shower doors. Smaller ones can ride in a
rabbet.
They won't be dust proof but will keep the
possums out, are cheap, and could use essentially
your original plan.

I want fast cheap and dimensionally stable.