A/C Will Not Blow Cold

Bought this car used and I'm having the above issue with the a/c. When I turn the system on, the compressor will cycle on and off at approximately a 5-6 second interval. I assumed it was low on freon and bought a recharge kit at the parts store. It reads around 50psi with the system on max connected to the low-side port. The line going into the firewall is nice and cold, but the air blowing inside the cabin is warm. If I turn the thermostat over to heat, it blows very hot, so I believe the blend door is working properly. Even if I short the pressure switch to cause the compressor to stay engaged, the line will ice up, but still only warm air inside the cabin.