Radio not working

The dash lights aren't a valid clue. Those are part of the instrument panel lighting circuit and work independently of the radio itself. The best suspect is the radio's memory fuse. That is always tied in with some other circuit that is always live such as the horn, cigarette lighter on some models, but most commonly the interior lights.

The fuse box is listed as, "Behind RH Side Of Instrument Panel, In Or Near Glove Box.".

Check the "Courtesy Lights: fuse with the red arrow pointing to it in this drawing.

https://www.2carpros.com/articles/how-to-check-a-car-fuse

Next would be to remove the radio, then measure the two power supply voltages. Best is to leave the connectors plugged in, and back-probe through the back of the plugs next to the wires. You should find 12 volts on the green wire all the time. 12 volts should be on the yellow wire only when the ignition switch is turned on.

Here's GM's radio removal instructions:

1985---88
1. Disconnect battery ground cable.
2. Loosen instrument panel insulator panel attaching screws, then remove steering column trim cover.
3. Remove ash tray, ash tray assembly and fuse block, separate fuse block and ash tray and push forward to gain access to cigar lighter and rear defogger switch electrical connectors.
4. Disconnect electrical connectors, then remove cigar lighter and glove box.
5. Remove instrument panel center trim panel retaining nuts, then pull trim panel outward to gain access to radio attaching screws.
6. Remove attaching screws, disconnect electrical and antenna connections, then remove radio from vehicle.
7. Reverse procedure to install.

I can't find any drawings to go with those instructions.

If both 12 volts are there, the problem is inside the radio. The ground should be okay since the back-lighting is working with the dash lights, but to be sure, check for 12 volts on the metal case of the radio. For all of these voltage checks, the meter's ground probe should be connected to the car's body or an unpainted metal support bracket that's bolted to the body. Don't use the radio's case for the ground.

Double-check those voltages any way just to be sure they're there. If one is missing, the new radio will also not work.

Dandy. Happy to hear you don't have to sing to yourself. I know my friends and neighbors are happy when I don't sing.