Motherboard screws have nothing at all to do with any possibility of shorting to the case (Or grounding for that matter, since they are isolated from the rest of the board) unless there is a standoff installed in the case under the board in a location that does not directly line up with one of the mounting holes in your motherboard. If there is, then that obviously is a big problem. If there is not, then this can't be a problem (Unless maybe you are using fasteners with extremely large heads that weren't meant for this usage, and probably not even then.) and you should tighten them back down. Look for the problem to be elsewhere, like having a standoff that doesn't belong, screw or other object trapped between the board and the case, etc. IF it is even related to this kind of thing, which it probably isn't since it was running fine previously.
I would HIGHLY recommend that you remove it ALL from the case, and using the box the motherboard came in (Or a piece of flat cardboard) try bench testing it using my guide.
Finding the problem through bench testing If you are here then it’s likely you have encountered a serious hardware issue and have been unable to resolve it using the standard no-POST troubleshooting procedures. If you have not yet attempted to resolve your issues using the no-POST...
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Obviously the power supply and motherboard are the primary culprits here, but bench testing might turn something else up. I also would not rule out the possibility of a bent pin on the motherboard under the CPU. We've seen a lot of systems with one or more bent pins that worked fine for a while, and then suddenly didn't anymore. It is always worth checking for that when none of the more obviously common problems turn anything up.