The fan is likely dirty. Remove the PS, clean it out. Peel the sticker off the fan, you'll see a plug. Remove it, and put a single drop of 3-in-1 oil in there (NOT WD-40). Reassemble and test.
And no, I don't think a scrambled picture is from your PS.
And no, you wouldn't resolder your monitor chassis while it's on and running. You must, of course, adjust it while it's on, but that shouldn't be a big deal.