I did a little test by running a game and checking the temperatures, which I realize isn't really the safest idea, though I have been running games despite this problem for a long time anyway. It got really hot back up to the 60s very fast, with the fan still on it. Also I had said alt-tabbing would make the temperature gauge on my tower go down and stop the beeping but there was no difference on the other programs. It didn't beep at all actually and was even cooler on the tower gauge than normal, even though the other programs kept showing the temperature going up even higher than 69C.
Obviously at least one of these senors is wrong, and I guess I'll trust two separate programs over some wacky one on my tower, though I'm not too excited about having to reapply thermal paste. Though it still seems pretty weird to me, these aren't brand new incredibly demanding games we're talking about here.