A/C compressor clutch not engaging?

Up on the cowl where the heat and AC get the fresh air from, the water will go down through the case drain and come out the bottom. Hopefully oil and dye comes out there if the core is where the leak is.

It will make the dye fluoresce so that you can see where the dye is. You still need to look over all the parts to find the dye though. This somewhat shows what the dye looks like when lit up. You can see how working in the dark makes it easier.
https://www.youtube.com/watch?v=4NnIarJjXUk

With the AC on and a humid day the drain will drop condensate, if there is a leak in there you can usually see a glow in the water.

That can be caused by dye that was in the system before or the dye reacted with the oil and glows a different color. Condenser leaks are a common thing on GMs, Use some brake clean and try to clean the existing material off. Then see if it returns. If so it's a leak. The evap cores are tough. Normally I will go over everything else very carefully. If I don't find something then I connect up a pressure gauge to one side of the core and apply air pressure to the other. Say I put in 100 psi. Then close the valve so that the air is held in the core and the gauge. If it stays at 100 psi for 20 minutes, there isn't a leak there. If however, the pressure starts dropping, there is a leak in the core. You can also do that with a vacuum pump, block off one side and use the pump and gauge to pull a vacuum and see if it holds. Same thing, if it holds a vacuum for 20 minutes it isn't leaking. Visual inspection can be done using a bore scope or similar. You snake it through the vents and look at the core, looking for dampness or corrosion. I didn't see a lot of Evap core failures on those, but condensers and compressors were common.

Unless you plan on starting an AC repair business you would be better off just hitting the local parts stores and using the loaner tools. That way you don't have money tied up in items that just set there. However you can search online easily enough.

I don't know of any good sniffers that are cheap, the one I use was $600.00 back when I got it with the $7,000.00 R1234YF machine. With it leaking that much the dye you put in it should be showing up by now. Did you check the leak you found earlier?

Nope. With an AC system you need to use items that don't leave residue behind as they will cause corrosion or acid formation in the system. That is why I normally use CO2 or Nitrogen as a test gas. They are "Dry" gases that don't cause problems and you can get detectors for them. They can also be used at a high enough pressure that the leaks will show up. They also work to test the compressor for operation because they will flow similar to the refrigerant, and if the system has a leak they are not illegal items to vent like refrigerants are.

Not directly. The pressure is used to push out the dye. You find the dye or use soapy water and look for the bubbles. You won't hear it either. Then once you repair it you need to let it stay under vacuum until all the moisture is out of the system.

You normally need a pretty good charge for a sniffer to work and generally you want the system working so you have pressure in it.
https://www.2carpros.com/articles/air-conditioner-leak-detection
https://www.2carpros.com/articles/replace-air-conditioner-compressor

You can replace the valve in it, you need the correct valve stem tool and the replacement stem. Then it unscrews from the port like a tire valve.