2002 Chevrolet Impala Back up lamps

We bought the car 3 days ago from a private owner who purchased it from the county. It was a police car and was well maintained.

We noticed that the back up lamps would come on, stay on for a short period of time, then go off. They would go dim and then brighten up and this is when the car is turned off. It eventually ran the battery down. It seemed
to come on at night when no one was around. We found out it stayed on because the neighbor came and told us.

When charging the battery up, they came on for a few seconds then went off and stayed off. I checked every couple hours and the lights didn't come on. By morning the battery was down. The neighbor said she saw them on at around 5 am and they stayed on steady. I did contact the previous owner and asked him if he had noticed that and he said no.

I have researched and not much shows up about those lights going on and off or just staying on. I did research about batteries and I did read that a bad battery could cause electrical problems.
The lights come on bright when in reverse, like normal. They just don't seem to want to stay off.

Any help would be appreciated. Thank you.