I think that the screws aren't as tight as you believe.
The screws must be slightly loose. If they're snug the neck won't move. Get a screwdriver with a good quality tip, align the neck properly and tighten them further.
If they keep turning, slack the strings, remove the neck, jam toothpicks and white glue into the screw holes, let it dry a bit, then re-install the neck and make sure the screws are tight.
"The best way to tell a lie is to tell the right amount of the truth, and then shut up." Robert A. Heinlein