A/C Compressor cycles, low pressure side is too high

From inside the car, here's what happens: When I turn the A/C on (car speed is not a factor), it starts out blowing cold for a second, then gradually gets warmer until it's about ambient temperature, then the compressor clicks off, air gets much warmer for a bit, then the cycle repeats. I would say when the compressor is on, it probably stays on for about 10-15 seconds each time.

From under the hood: I have a gauge for the LP side that came off of an OTC recharge can, and here is what I found.
First, a bit of context: a while back, I was having this same problem; I got the gauge out and the LP side was showing on the low end of the optimal range, so I thought maybe it just needs more refrigerant. I added a can, the reading looked better afterward, and the operation seemed to improve a bit.
Back to today: I connected the gauge with the engine running and AC off, and I noticed it was hard to push the connector on. When I finally got it on, the reading explained why I had trouble: it was off the charts! (The gauge needle went all the way around to the back side of the stop pin.) Anyway, I left the gauge connected and turned the AC on. The compressor engaged, and pressure dropped to around 60 psi, fluctuated for a few seconds, and then started to climb. As it did so, the air temperature at the vents climbed with it. When the pressure got up to around 80 or so, the compressor clicked off, the pressure jumped up to 90+ and creeped upward from there. (My gauge maxes out at 100 but the needle can go past that.) When the needle got up to about 125-ish, the compressor clicked back on and the cycle repeated.

I feel like this amount of detail should probably give an experienced tech a decent idea of what the culprit might be, so I'm wondering what could be causing the LP side to have such high pressure. Unfortunately, I don't have a gauge for the HP side. And I have only added that one can of refrigerant the last time this happened. Definitely not enough to bring the pressure up that much.

Both radiator fans are operating normally.