Saturn ion 2004 AC problem

My 2004 Saturn Ion, 235,000 miles, 2.2 engine is idling erratic when I turn on AC. It sounds as though the compressor is turning on and off. The air is cold then blows warm then cold etc. Is it the compressor or could it be a relay or fuse. It also overheated last week but not sure if problems are connected. I am idling car right now with AC off to see if fan comes on. Any thoughts?

I ran car without AC for about an hour and the temp started to go in the red so I am pretty sure I have 2 departed issues.

Do a pressure leakdown test and post back with the results.

sorry I haven’t replied sooner. I do not have tools to do that check. I think I may have to take car to shop for a/c problem. I did however find the cooling fan is not coming on. I wired directly to power and it did not run. found 1 on ebay for $85.