A/C compressor clutch not engaging?

Normally you just put some in each. Then set the compressor on the pulley. Wait a few minutes and rotate the compressor by hand with it horizontal. That allows the oil to move around.

Well, if they say it was filled I would almost trust them. They are the ones who have to warranty it if they didn't. The pulley will turn free, the front plate is what turns the compressor, that is what you need to rotate, it will be stiff but it should turn with moderate force.

Orifice tube goes in the high pressure line (smaller line) between the condenser core and the evaporator core in the dash. Be sure it goes in the same way the current one comes out.

This is about the only image that shows the lines.

Yep you have a leak. Is this after the compressor replacement? If so then start looking along the lines for oil or discoloration, Then look real close at the condenser core up front, It doesn't take a lot to cause them to leak. If you don't find it there then the next step is to use a sniffer and put a small charge in the system and see what the sniffer shows. Some parts stores have them as loaner tools. However they are somewhat fussy to use. Normally you want no moving air and you will need to hold the detector tip close for it to find the leak. The other option is soapy water. For that you use an inert gas like CO2 or argon, you put 100 psi in the system and then start spraying or mopping the connections and lines then the cores and see where the bubbles show up.

You can put dye in the system, but you need some pressure in there to push it out of the leak. Without that the dye will just lay there. You could try a can of 134 with dye in it and see if you find anything.

DON'T use any type of sealant, 99% of them don't work, but they do cause problems in the cores and compressors and if you use it no shop will ever touch that AC system again because the stuff destroys the filters and lines on the machines.
As for what to use, I use CO2 and Argon, use both for welding as well. For the DIY you normally would add the r134a with dye in it, knowing you will see it leak out and the money is gone. That is what most of the sniffers look for as well. I've also used regular shop air if you have a compressor that doesn't dump oil in the lines. 90 psi does good, then you run a long vacuum on it to be sure you get the moisture back out.

Maybe, depends on where you bought it. Tell them you grabbed the wrong one. To use air I made a line using a fitting and a coupler for the high side and used that.

You are looking for a leak, put in one, then run the engine and jump the compressor if you need to. Let it run a bit, then shut it off and use a UV in a dark room to look for the leaks. Pour a little water into the fresh air intake to see if dye comes out the HVAC drain. It can be tricky. The adapter is nothing more than what you have if you took a low side coupler and line, and used an adapter that fits your compressors coupler.

I've tossed a blanket over the area and used that.