I usually just play my acoustic but I have a cheap electric that I was recently reunited with that I want to play. Problem is the intonation is in a ridiculous state.

I messed about with the screws on the bridge but even when they're as tight as they can go or as loose as they can go, the situation barely changes.

I figure either the bridge needs to be moved up or down the body (does that mean drilling new holes?), or maybe it can be fixed with truss rod adjustment. What do you think?
go to the setup thread and read up on intonation. Learn to do it properly before you destroy your guitar.
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I know how to do it properly. I even did a music tech course recently. But if the 12th fret note is massively flat when the screw is really tight, and massively flat when the screw is really lose, obviously it's gonna take more than screw adjustments.