My toilet seat shifts from side to side. The problem seems to be that one of the screws is loose in the hole provided in the porcelain. Could I fill the hole with caulk or liquid nails or something like that then drive the screw into that filler?
Well, in most cases, the bolt isn't designed to completely-fill the width of the hole. Rather, the wiggling is prevented by the bolt and nut being tightened down, on which side depends upon the design of the hardware. I would that whatever the appropriate tools are and tighten the thing up. Straighten it out like you like, then tighten both sides.
There are enough designs that it's hard to generalize too much, but that's the basic idea.