1998 Chevrolet Corvette Electrical

To start with there's two things to look at. If you have a trailer hitch, check for wires that are chewed up and touching. Next, part of what you described is commonly caused by having the wrong bulbs installed. This shouldn't be an issue with a '98 model unless they're still using the 1157 bulbs with the round brass base. If you have those, but there's an 1156 installed, those just have one terminal in the center. 1157s have two terminals. One is for the brake light and one is for the tail / running light. An 1156 terminal will short the two contacts in the socket together.

A broken ground wire will cause similar symptoms. The clue there is if you remove all the bulbs in the rear, then press the brake pedal, the front ones won't turn on. If you install a rear bulb, then the front ones turn on with the brake pedal, we have to look for a bad ground.