A/C panel lights and compressor shutting off randomly?

Hello, I recently started having an issue with my A/C panel lights on the dash not coming on when I would first start my car. It starts off blowing hot air, so the fan is still blowing strong, but no cold air. Then after about a minute the lights on the dash panel suddenly come back on and the AC compressor clicks on, and it blows very cold air like it should.

This happened for about a week, but only when I would first start the car, and even after more than an hour of continuous driving, it would keep blowing perfectly cold air without interruption.

Just this week it started shutting off after I had been driving for a while. Again, the fan keeps blowing, but the /AC panel lights on the dash suddenly shut off, and it starts blowing warm air. It usually comes back on within a minute or so and the system goes right back to normal/cold air. Last night, for the first time, the A/C shut off and back on about 4 times within 10-15 seconds.

The strange thing about the lights on the panel all shutting off is that even if the A/C is shut off, the fan is still blowing, so the lights on the dash for "re-circulate" and for the fan location control (face, feet, blended, etc, ) should still be on, since the fan is still blowing strong, but all lights shut off, not just the A/C light.

Since the A/C is able to blow very cold, and for long periods of time (I have a 1hour 15 minutes long commute), I don't think it is a refrigerant pressure issue. And since all the AC and fan lights are shutting, even though the fan keeps running.I'm wondering if it is something like a relay that is slowing failing, but I'm not sure if that could be the case, and don't know how I would diagnose that.

Thanks for your help!