Not that?! Okay, so, now we're getting to the slightly more destructive options.

Do you feel it wigging, getting loose but not quite, anything like that? Okay, get your craft stick (or whatever you chose). Deposit your cotton/absorbent material or Scotch tape around the screw to avoid accidentally gluing your screw into the socket (and thereby worsening your situation, BADLY...)

Put a SMALL AMOUNT of the strong adhesive on the head of the screw and wait for it to dry. Periodically check up on it in case it's leaking over the side. Then, once it's bound (might take a good long while), yank it out. This might get the screw to just rip out caveman style. It's not elegant but it might work.