Identify screw and dowel on this cabinet door

Might not be the answer you're looking for, but (as was the answer to that other question) you could pull it out and get a new plug. You might not need to redrill if you're careful or get a plug that'll hook into whatever hole is left.

Also, I know it could just be preference, but have you tried Gorilla Glue?

I agree with ArchonOSX, you need to pull on the screw as you screw it in. But, be careful if isn't going deep enough. You could be hitting a wire or plumbing or the screw may just be too long for a framing piece.

If that's the case you need to remove the first screw & start over with a shortened screw. Thread the wing on past where you want to cut or snap the screw end off. Cut or snap the screw end off & you're ready to install.