My Carrier outside condenser unit continues to run after the residence thermostat has shut down the system and the furnace blower fan is stopped. I suspect a relay failure on this split system. Can I correct this problem myself or should a professional be called?

Likely you have correctly identified the problem. No big deal to change it for anyone with the sense to shut the power off. Should be a disconnect right at the unit. Might short the hot side of the relay contacts to discharge the big capacitors used in starting the unit. Use the blade of a plastic handled screwdriver to contact the contacts and the frame of the unit. Then remove the relay and take it to an electrical supply place. The more consumer friendly home centers may have them too. 24 volt coils are common. From there, make sure the amps are as high, or higher than on the unit name plate.