Okay...I just had to find out what the deal was.
Got the wife's 1997 Camry XLE, V6, automatic and tried to reproduce the same problem. Dash lights and headlights work. Brake lights and hazards work. Turn signals work. Dome light works. Corner and tail lights work.
Located the Tail fuse in the interior fuse box.
The one at the bottom of the middle row--red 10A fuse. I pulled the fuse. I checked the corner lights as well as the tail lights. Nothing. No lights at all. I could hear the relay (located somewhere behind? the center console) click when I turned the light switch on/off. I put the fuse back in and tried the switch again. It works!
If it's not the fuse...and not the sockets...and you replaced the switch and the relay...not much else left except to trace the wire and see where the break is.