The fan nor compressor will start up and the buzzing is certainly coming from the contactor as I can feel the vibration and hear the noise from that part.

The YouTube videos I watched were pretty clear it was the startup capacitor so I replaced it and the dual capacitor.
I made sure to solder the 15k ohm resistor back on the new startup cap and the leads are firmly connected and in the correct spots.
The fan and compressor make no tiny motions or sounds at all.
The contactor buzzing noise seems much quieter than before I replaced the caps.
The buzzing is continuous but I don't allow it to go on for more than 30-60 seconds.

So the problem is actually the contactor itself or the extra dangling capacitor that's on my unit, that's not in the wiring diagram, or the potential relay or the compressor itself?

Maybe I should pull the extra cap and see if these two brand new caps are enough to start up the unit.
Why would the extra cap be added? It's connected to a slender wire running back up the freon line into the house to the cutoff switch on the side of the furnace.

EDIT1: I tried holding down the contactor for a manual start and it resisted and buzzed very loudly.
EDIT2: Ahha, I've inspected the wiring several times but I missed the chew marks at the very top of the wires running from that extra cap to the cooling fan. There is s till some copper but maybe not enough.
I'll have to use the ohm meter and check every wire even though I don't see any chew marks on the rest.
Must been a mouse.

You could turn power off to compressor, remove load wires(compressor/fan), turn unit on, see if contactor engages and is relatively quiet. Verify 24 volts to contactor coil, and 240 to contactor Line side.
Turn power back off, disconnect compressor, try again. I am thinking it may be the compressor. Remove power again, test compressor leads for a short to ground.
Should be detailed videos on YouTube.
Good luck, let us know what you find. Thanks

There was only 1V of power coming from the main line. I had pulled the main breaker plate when working on it ans assumed (wrongly) that I had 240 V.Resetting the 30 breaker several times and finally opened up the load box to find an insidious problem.The weather cowling over the incoming home electric lines was missing for years and rain had dripped into the breaker box.The 30 amp breaker is corroded and shot and I'm going to have to replace the entire load panel.The weather cowling is replaced but the original owners never had Ameren install a cutoff switch so will have ask them to install one or have the power cutoff for a day while I replace the box. Never done this before and am watching training videos and ordering an Eaton 100 amp unit from Home Depot. I'll have two 30 amp and 8x 15 amp breakers. All incoming lines to the house will have their own breaker now. No more washer/dryer/fridge/microwave/coffee maker on one 15amp popping if you use more than 2 each at a time.It'll be a week before it gets here and 100 degrees most days... At least the attic fan and window fans work. The window A/C unit pops the 15amp breaker in 3 minutes. I'd like to wring the neck of the person that wired up this little house.