Replaced the motor in our Coleman Mach 15 with a FASCO from Amazon. Does yours look like this?
If so:
1. It's pretty easy to change. On ours, it was pretty easy to change. I got good at it. See 2. below... The Inside Blower (squirrel cage), the plate that attaches Motor to Blower Housing, Motor, and Outside Fan, all come out as one piece. Our Fan was retained to the shaft with a hex screw clam. The Blower uses a 3/16" Hex Wrench (Allen Key) and if you have a long-enough one, there's a spot in the fins that gives you a straight shot at the Screw.
2. I had ours out several times before I ordered the above
from Amazon here.Several times because the Sleeve Bearings got sticky once hot and wouldn't let the motor re-start. I can only think the porous material got soaked with something that kept keeping coming up sticky. So...
I kept that motor after installing the new one. The windings are still good. If you want parts, PM me.
I'd suspect something is in fact wrong with the windings in yours. When you tried the known good Fan Run Cap, did you swap the one from your suspect Motor into the Other A/C and confirm that it actually ran the other Motor? As well as the other one running the suspect Motor?
Around $100 for a Motor is good, especially if you can avoid a service call.