2000 Camry LE 4 cyl. Both brake lights suddenly went out but the back windshield light still works. I replaced both bulbs, checked the fuse and checked the brake pedal switch. The switch contacts showed continuity on one set and open on the other. The result reversed when the switch was operated. They still don't work.

I believe you have some electrical wiring problems in the trunk area.
See the link to this post I made a while back. This is quite common on brake light, and other electrical gremlins.
https://www.askmehelpdesk.com/cars-trucks/2000-camry-blowing-fuse-433155.html#post2168051

Hey guys! I had same problem with my 97 Camry last week. I replaced the bulbs 3 times but they kept exploding. I finally removed the bulbs and just didn't worry about no brake lights - until I got pulled over by the po-po. So last night I glued 2 red bicycle reflectors to the lenses! Now lets see if the po-po even notices. As soon as he hits his lights to pull me - the reflectors should glow bright red and he can't prove nuthin on me! Ive caught my last ticket!! BTW Western Auto has the best price on reflectors. I shopped around until I get the best deal.