Homework, you did the right thing to come here. You will get good wisdom from these guys.
In this case, your handyman made an obvious mistake. He might not be the best at this. "It tested live at the circuit breaker" is silly --- and a clear sign. ALL the breakers test live !!
Get an electrician to restore power to the wires he cut at the breaker. You might have to dig out the old wiring to see if by re-energizing those wires, the wires in the wall get hot again. They should NOT go hot.
Why? Heavy drain items like HVAC require dedicated circuits. The old HVAC unit and your existing should NOT be on the same breaker. What probably happened was the old unit was abandoned and the wiring re-routed but the plug remained, disconnected at some junction box.
TURN OFF POWER. (May seem redundant but get in the habit) Look around your existing HVAC for this box. You will probably find the old wall wiring there.