Does the compressor stop? Turn it on, listen/feel if motor slows a bit. If it picks up when the A/C stops blowing cold the switch or relay aren't is not holding. The fact it blows cold so soon makes me think it has freon. Possibility the belt is loose? Loose connection to compressor clutch?
It's also possible low oil is affecting the compressor so it stops when it starts to build up pressure (load). See it it turns freely by hand with switch on but motor off. It should move with effort but not stick.
Getting to the switch means finding the relay and/or removing dash. If compressor loading down, it needs a pro to service it. You can check the belt yourself.