Thought I would chime in with my 2 cents worth...
You said earlier you didn't feel the same magnetic pull on one motor when you turned it by hand. Since the motors have permanent magnets that do not need electricity to energize, you would expect them to all feel very similar. They . That being the case, simply replace the one that feels different. You do realize the motors are only $18 retail.
As FlyNfrank said regarding soldering problems, just cut the wires and solder in the middle of the wire run instead of attempting soldering to the board. If the wires are twisted together before soldering they won't fail.
Just do it and tell us how it comes out. This is becoming too painful to read without you moving on to the ultimate fix.