Thanks for the input folks!
Stand_watie and 280plus, I have tried with new tapes and the problem does seem to be reduced but not eliminated. My theory is that since the tape is on the spool and is laid flat on it's side in the tape deck that the bouncing around in the car kind of knocks some of the tape out of perfect alignment, Chinese yo-yo style. That way, even if your tape is new, it will eventually get harder for the tape to play if you listen to it in the car. Then you run into the "auto-reverse/anti-tape eating" issue.
Actually, now that you mention it Greg, I had the same problem with the little rubber bands drying out in my last car, a Pontiac (or so a friend suggested). That car was old and fixing the tape deck would have been like rearranging the deck chairs on the Atlantic so I listened to tapes using a small battery operated tape player. The Camry however is a good car and I use it a lot for work so I might start looking for a new tape deck or maybe even a CD player.
Thanks again for the help. I guess I'll be putting a new gadget on the wishlist.