AC problem. Compressor not running

Make sure that your drip pan is not full of water. You have a switch in the pan that will turn things off if water gets in it. Clean out your drip tube as well with bleach monthly, especially in the heat of the summer.

Could be the capacitor - dual run capacitor is in essence 2 capacitors in a single unit, so one side can be functional, I.e., fan in your case, and the other side, the herm (=compressor) can be bad, or vice versa. You have a multimeter with a mF function (microfarad) - easily checked. Watch a couple YouTube videos.

Have you considered the high pressure cutout switch? Just had some service on mine and it was cutting off due to high pressures (there was restrictions in the line, thus the call to the tech). I watched him as he bypassed that switch to get the compressor to turn on but that was in diagnostics, I don't recommend it.

You could have a bad switch or something else making the pressures high.