One quick diagnostic practice is to take your screwdriver and push in on the center of the contactor and see if the compressor comes on. Also, while you're at the contactor, locate two small wires on the side of it and check them for 24v. Those wires activate a magnetic coil to pull the contactor in.
Now, turn the power off and take the little hood off the side of the compressor where the wires go in and see if all the wires are still connected; they are bad about burning off over the years. Next, take all the wires off the compressor and find three terminals marked S, R and C; take a digital ohmmeter and check from C to S, then C to R, then S to R. The first two readings added together should equal the third reading. If that is not true, the compressor is burned out.
Let us know how you come out and if we can help any farther.