Tried to look up the ratings of the Capacitor. I can tell you the FAN side is "10" which is 10-mfd, 10uF, or however it's expressed. Ten MicroFarads. You can see the "10" below "M" in "A.O. SMITH" in one of the photos.
Would be best to loosen the strap around the Capacitor and read the uF value off of it. Might be 35, might not. You can get a single capacitor with the correct value for the Compressor (might be marked COMP or HERM) and disconnect HERM from the combination Capacitor you have now. Connect that to the single value one, but you'd need to run a jumper between COM on the existing Capacitor and the other terminal on the new single one.
It also has a Start Assist Kit on it. That little black capacitor is probably 88-108uF, and there's a PTCR device with it. You can see those in the wiring diagram. You shouldn't need those in circuit to start the compressor on 30A shore tie, but the HERM side of the Capacitor has to be there and be good.
I'm good at this part of the A/C, not so much on Controls and the Control Board. Compressors fail, but not as often as you'd think. Capacitors fail often enough that I keep spares on board. But I'm not much help on the controls. Fortunately for me, our A/C (Coleman Mach) is not a ducted unit, so it doesn't have a wall thermostat, control board etc.
Anyhow, I've got this nagging feeling your problem is in the controls. Still worth it to check/substitute the capacitor. Some meters can test capacitors but you've gotta get a little upscale from the Harbor Fright freebie and the like.
