Oy vay - that is a drag.
THere are a couple of tools for getting buttons out - some vintage, some new ones. THe new ones aren't cheap($100-125).
My suggestion would be to leave your sticky buttons in the case. First try cleaning out with some 90% isopropyl and then lubricating with a super light oil (CRC's #2-26 "electrical grade is what I prefer, found at Home Depots in electrical). BUt if your button "hangs" despite otherwise smooth movement, it is more likely it has a ding on the button or the edge of the case where button goes in. Those can be troublesome - if you can see the spot, you can push a piece of rubber against the opposite side so it doesn't spin(good way to smoke the button and end up where you are) and you can polish or sand the ding. I'd suggest 1500 grade paper, sold wherever they sell auto body finishing products. Good luck - or just continue the thread.