I'm familiar with Coleman, but I'd still guess that there are several capacitors in a Carrier A/C unit:
1. Fan Run (smallish metal "can")
2. Compressor Run (larger metal "can")
BUT... 1. and 2. might be combined in one larger "can" with terminals marked "Fan" and "Herm" for Hermetically Sealed, the Compressor part)
3. (probably) Compressor Start (very likely a plastic case) and some sort of Start Device, Relay or special Resistor)
Sounds like Fan runs OK and the issue's with Compressor. Some systems have a delay built in, so Fan and Compressor don't start at the same time. That could explain some of the strangeness you're experiencing.
After a few unsuccessful start attempts, the compressor will get hot enough that a thermal safety opens the circuit and it won't try to start again till it cools, which could be 15 minutes, maybe more.
Are your Front and Rear A/C units alike? If so, you could switch the capacitors between them and see if the problem stays with the unit or follows a capacitor.
If you want to try to swap them, open the circuit breakers before you take the plastic cover and the metal shroud around the capacitors. Then before you work on the capacitors, short across the terminals to be sure they're discharged. Draw a wiring diagram and/or take pictures. If you can get pictures, please photo the capacitors, and the wiring diagram if there's one stuck to the shrouding up there. Post'em here.