The
Electronics repair FAQ is great. Almost everything can be diagnosed with just an ohm meter, but sometimes the component first has to be removed from the circuit. If the fuse hasn't blown, then maybe the damage is on the low voltage side -- measure each of the big diodes on the heatsink. If the fuse has blown, then the high voltage transistors may have shorted. Replacement parts rarely have to be 100% identical to the originals, just of the same general type and rated for at least as much voltage, current, and power. One exception is when two diodes or two transistors are in parallel, in which case both replacements have to be 100% identical to one another (can be different from the originals).
A soldering gun is a good way to overheat everything. A 40W soldering iron is a much safer choice.
I'd rather use heatshrink tubing than electrical tape over wires because it doesn't unravel.
Get a 100% plastic screwdriver or carve a chopstick or wooden dowel into the shape of a screwdriver blade to adjust the pots because high voltage is all over inside the PSU, even where you don't always expect it, like the big heatsinks. Read the precautions in that repair FAQ, and if you absolutely must work on a PSU while it's plugged in, be sure that the wall outlet is both grounded (some 3-prong sockets aren't really grounded) AND protected by a ground fault interrupter, but those things won't prevent all injuries.
One thing you don't want to do is bypass the overvoltage protection!
I would check if the pots have any sealant on them that could help restore them to their original positions. If that's not possible, I'd set each pot to its center and then tweak them if necessary while the PSU is under a roughly 50% load. Don't adjust a PSU without a load or the voltages could end up too low (sometimes too high). Don't use a mobo for the load because it could get fried. I'd adjust the +3.3V last (probably on a tiny circuitboard, right where the wires exit the PSU, probably next to some orange wires). Probably the first pot to adjust is the one near the middle, on the other side of the heatsink.