Hobart P660 keeps shutting off and tripping the breaker

Ran into this many times. It more than likely the mechanical starting switch is stuck so the start windings are staying in constantly instead of momentarily engaged. There's 2 ways to tackle this problem, the old fashioned way which reqires motor removal and diss-assembly to replace the switch OR with a little bit of ingenuity you can install an external Sinpac electronic start switch. A company called Carbon Brush sells them and can E-mail you the wiring diagram. Good Luck!

Tripping the breaker is probably a direct short circuit. It sounds like the initial problem was with the contactor but running equipment with a "chattering" contactor can cause damage to other electrical components, as you found with the capacitors.

The firs thing I would check is the coil on the contactor. Make sure the coil was rated for the correct voltage. It is a common mistake for someone to install a 120v contactor with a 24v coil when you need a 120v coil. It would run for a little bit then burn up (possibly creating the direct short). If you have an OHM or conductivity meter (usually a setting on a muti-meter or "voltmeter") you can disconnect the power and check across the coil to see if it shorted. If it's the wrong voltage coil that's likely the problem. Next you can check the motor for a short in the same way. You can also see if you can turn the mechanism by hand. If it is locked up, it may trip the breaker.

There could be many other things that would cause it. A burnt wire the shorted to something grounded, any other power consuming component could cause a short.

Sounds like the cable is too small. If you're hooking it up to 15 amp breaker you need the correct diameter size extension cord, and the equipment cannot be rated more than 15 amp or you'll have a problem. I perceive this may be the solution based on the fact that you said that the motor was not coming up to speed.