Sometimes the reverse light switch can work it's way loose, and slip a bit, causing the reverse lights to be on all the time. If you have one bad bulb as well, this would explain everything.
Have someone stand behind the car with the key to "aux power" (not engine running) then reach under the car (at the driver side) to the switch on the bottom of the trans (green and red wires going to it). Make sure the car will not roll away (but not in reverse). Then try to rotate the switch assembly clock wise and counter clockwise. If the lights go out. . .tighten the two screws. then check to see that the lights some on in reverse.