Something got bent/tweaked/out of adjustment down at the linkage on the trans. Your reverse lights go on when the shifter is in reverse because the switch is at the shifter, not the trans. It's actually a combination neutral safety/gear range sensor switch. The shifter doesn't know what gear the trans is in, only what gear it's supposed to be in. Disconnect the cable at the trans linkage arm, rotate the lever by hand to put it into park, then adjust your cable as necessary to get it to line up with the linkage arm.

 

Thanks for your help Camaro 69. When I replaced the shifter cable, I did put the transmisssion and shifter in the Park position before re-attaching the shifter cable, but it does seem to still be off by one position. Should I disconnect the shift lever arm on the transmission and then realign with the shifter cable? Also, if the problem is just improper alignment of the shifter/shift cable, shouldn't the car actually be in Reverse, when the shifter is in Park, and therefore move, since the shifter seems to be off by one position? Or, does putting the shifter into Park, electronically disable the Reverse gear?

Again, thanks for your help.

 

i had a similar issue years ago. the bolts for the bracket magicly loosened after i had the car in the shop for tire work. it acted exactly as you described. how long have you owned the car? has it progressively gotten worse, or did it just happen one day?

 

The car was given to me, mostly in pieces. The transmission problem was there when I got it. I've been busy putting it back together and getting it ready for painting, and just now got around to the transmission issue. I'm sure the bracket is secure, as I removed and replaced it when I put the new shifter cable on it.