Cusser
+1y
My first thoughts would be to check the fan clutch for the radiator/condenser fan. Do a Google search on how to do that. Maybe bad fan clutch is causing high pressures and safety shut off. What you hear is the AC compressor clutch engaging, the magnetic clutch pulls in the hub so it starts pumping your refrigerant.
Second guess might be to measure/adjust the clutch gap.
Third would be to have a real AC shop measure refrigerant pressures, and recover and weigh the refrigerant, only real way to find out if there's been a refrigerant leak, as a leak and low pressures could also be a big part here.
I really think a REAL AC shop could readily troubleshoot this, AC work takes special tools and knowledge.