The fan that I replaced was merely the case cooling fan. Not the cpu cooler, not the power supply fan.
I plugged the new fan into a connector to make sure it worked. 12 volts is not that scary. I have seen electricians work with live wires at 110 and 230 without a problem as long as you are not grounded and work with insulated tools.
Yes, I should have turned off the computer before cutting the wires to the old fan, but I was involved in a conversation and was distracted. Even so, had I been paying enough attention to cut one wire at a time there would have been no problem, but cutting both positive and negative at once can complete the circuit through the cutting blade.
I yanked the psu out of a leftover and the computer worked fine. Out of curiousity I decided to try the original psu once more before tossing it (because shorting a peripheral circuit really should not ruin a psu) and the computer worked fine again.
I don't understand why the computer would not work for a while, but at least it works now.
Obviously, I realized as soon as I made the cut that I should have turned off the computer. I posted my question because I know that I am not the only person who has ever made a foolish mistake. I thought someone might be able to help me.