Two areas of concern:
1). It's possible you've got too much refrigerant in there and it's messing up the pressure bad. What makes me think that is the ten minute delay. I believe that as air passes through the condenser the refrigant cools down and the pressure reduces enough to allow clutch engagement.
The only way to know for sure is to conduct a performance test. You need stopwatch for clutch on off times, pressure gauges on both sides of compressor (suction and discharge hoses) and a thermometer. You will run the car at a given rpm. Max A/C mode full blower. Match your output against test standard for you make / model of vehicle at given ambient temperature. You will need the service manual for your car to know the right pressures for both suction and discharge gauges.
2). It's also possible the air conditioning clutch is not working correctly. That plate sits on three flat springs and an electromagnet coil pulls plate to spinning pulley to start the compressor spinning. It's possible one or two of those springs is broken.
Another critical component of the clutch is the air gap between the plate and the pulley. That characteristic is critical. If the gap is too big the plate slips (and gets way hot). If the gap is too small the plate sometimes gets stuck on the spinning pulley and won't disengage. With car off and keys safely in your pocket take feeler gauges and measure that gap. I think the spec is 0.7 to 1mm but I'm not positive. Check your service manual for the right specification for your vehicle. Apparently there are many different air gaps used depending on A/C compressor. The air gap is set with a stack of shims / washers under the clutch plate mounting hub at the compressor main shaft.
Strong recommendation for you is to buy the Chilton or Haynes service manual for your vehicle. $30 well spent.