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Not sure for certain as the circuit should only be powered when the ignition is live
Ther reversing light swiitch on the transmission is notoriously unreliable. Often it causes fuses to blow. You could try changing that to see if it solves the problem. Its an easy change.
I've had trouble with these on the Camaro but not the Vette. In that case it was an internal breakdown of the switch causing the fuse to blow everry time reverse was selected. I guess it could cause similar symptoms to yours
It may be worth a pre-emptive change. The part is less than $20 so no big deal if it doesn't fix it. Disconnect the electrical connector and unscrew the old switch. Be ready with the new one as you'll get some fluid leaking as you change it.