I think pieces of corkboard with stickum on the back can be found in craft stores. Or just get some plain corkboard and glue it on.

especially if you're using it as a coaster and it's getting wet a lot. Water is the most destructive force of erosion.
You can try sealing it with a penetrating sealer, or you can look for a different kind of rock like dark marble or granite.

but i would check out a tile store for some 4 x 4 polished tiles. most places will let you buy just a few pieces.
granite fabricators often cut scrap into small tiles. some also give away scrap.
teflon glides are the new felt.

I have a pretty good sized chunk of petrified wood that I put about 5 good coats of Deft on. it isn't super crumbly but it holds together nicely, and gives it a deep glossy coating. Deft aerosol is very quick drying.