Any Ideas on Engine Starting Problem?

[musclecar fan]

It's a stock car with 350 engine, auto and stock cast exhaust manifolds, etc. I do have spring ring cables and have made sure they are clean and tight. I also took the ground screw out on the radiator support on the passenger side that attaches the ground wires and used emory cloth to make a good metal-to-metal connection. As to the copper ground cables between the engine block and firewall and frame they are all there and securely attached. When I turn the key the "single click sound" seems to originate under the dash and the batt and generator lights light up (red)as I turn the key. No overheating problems, etc and the battery is not going dead and remains fully charged. It's kind of embarrasing to drive it to a car show and not be able to move it after I shut it off. This is frustrating....