You are talking about blue/ run (not the brown/ bypass/ IGN2?) And this is right at the switch? If so that is LOW IF you still show 12.X TO the switch.
It is important to compare, ..........if the supply TO the switch has a problem, ANY load drawn on the switch (when on) will pull down supply voltage to the switch.
So measure again, just to be sure..........the voltage TO the switch / key off
Turn key on and see if that voltage DROPS. It might drop a few tenths, should not be much. If it drops a lot, follow the path from the key back out to the battery............welded splice, ammeter, bulkhead connector...........fuse link etc
