Ordered a 30 amp contactor and a 40 amp contactor, because I couldn't tell by the one that was on there which one I needed.
Short answer: it is running, and you're awesome.
Long answer (if you care): When the knucklehead that came and looked at it last was here, and charged me $145 to tell me that there wasn't power to the outside unit, I took apart the flexible conduit and found that indeed, both hots had melted through or otherwise corroded. I assumed this happened because both were loose and twisting around in the current contactor. I took the time to put an outside box on the wall, put in a 30 amp breaker, run the old wires to the box, and new wires from the box to the unit. The guy told me part of the problem was that the breaker in the house was 50 amps and the unit only needed 30 "and that extra power probably blew it."
So it worked just fine with the 30 amp breaker outside until our recent problem. Tonight I put in the 30 amp contactor because, again, I didn't know which to use. Put it together, shorted the wires behind the thermostat, and the unit turned on briefly before the breaker OUTSIDE (30 amp) flipped.
The contactor that had been in the unit was 40 amps, not 30 amps.
So I took out the breaker outside and just wire-nutted (technical term) the red to the red and the black to the black, then replaced the 30 amp contactor with the 40 amp contactor. Thermostat back on, flip to cool, turn the temp down, and VOILA!
It's been running for 20 minutes now. Once this 90+ degree heat goes away I'll probably put a shutoff back outside, but do 40 amps instead of 30. For now, I'm just going to let 'er run.
Thanks again so much for all of your help. And yes, swapping out the contactor was a piece of cake.
You rock, Billy. My family and I thank you.