If I had to go through all the trouble of shortening cables, I wouldn't make splices in the middle of wires but instead would either crimp new metal ends to the shortened wires (if a modular PSU) or solder them directly to the PSU circuit board (non-modular). If you try the latter, do not plug in the AC power cord without the PSU being fully reassembled, including its cover completely fastened in place, as there's way too much exposed high voltage in a PSU for safe operation often even on the big heatsinks.
Use rosin core solder that's either 60% tin, 40% lead (the most common type for electronics work) or 63% tin, 37% lead (slightly better). Avoid lead-free solder because it needs much higher temperature, and some types make the joints always look bad, meaning you can't tell whether they're good or bad. Try to make your solder splices so no sharp strands or solder needles stick out, which can pierce through any heatshrink insulation and cause a short. Cover the splice with 2 layers of heatshrink, each at least 1.5" longer than the splice. You want good mechanical reinforcement because soldered stranded wire becomes brittle, and PVC heatshrink is the stiffest. Most heatshrink is instead polysulfone, which is the softest (but is available in a stiff type). Teflon is in between. The only drawback to PVC is that you can't shrink it with the barrel or tip of a soldering iron because it will melt.
Don't cut all the wires of a cable in a single chomp. Instead cut one wire and splice it. Then cut another wire so its splice is 2" away from that splice. By staggering the splices apart you avoid creating a fat lump in one spot of the cable. You're by doing just one splice at a time, you're less likely to mix up wires, especially if they're all the same color.
There's also something called "Solder Splice", made just for splicing wires. It's clear heatshrink tubing with a ring of solder in the middle: (how-to video -- skip to 00:30)
. You can use a soldering iron instead of a heat gun, but a hair dryer won't get hot enough.
Before reconnecting the PSU to the computer, measure the voltage of each pin with a multimeter, in case any wires were crossed, especially if they're all one color (many modular Corsairs) or not standard colors.