1997 Toyota Camry Signal & Tail Lights Flicker for no r

Toyota Camry 1997 LE

Just bought this car used:

Tail lights and signal lights turn on and off while driving, when first starting and when A/C is on full blast. I hear a clickin sound in the dash while the lights turn on and off.
At times when I am getting out of the car, and I turn off the headlights, the signal lights and tail lights will not turn off. Sometimes it takes restarting the car, turning the headlights on and off several times, just so the signal and tail lights will finally turn off.

My battery is good, but I took it to AutoZone, and they determined that it has a hole in it. They said that could cause the problem.

Also, I'm not sure if the alarm system on the car is factory or aftermarket. Either way, I don't have the keyless entry remote.