I agree that the car is not the problem. I'm betting the CB is not internally fused and is not connected thru the fuse block. This causes any CB problem to effect the running of the car.
To track down your CB problem, connect your CB power wire through an ammeter and the antenna wire to a dummy load. Operate the radio without the car running. Try operating in the receive mode and transmit mode.
If the current draw is normal, take the ammeter out of the circuit and look for a problem with the antenna.
If the current draw is high (or zero), the problem is with the CB itself.