So let's break this problem down.
1. Reverse lights stay on:
This sounds like to me a issue with your stop lamp switch or transmission gear selector. The stop lamp switch tells your brake lights or reverse lights when to come on based on what gear you are in and if you are braking or not. It could be a sensor in your shiftier that is not telling your reverse lights that you are not in reverse!
2. Oil light brake light dash board lights (Christmas tree as we call it because it it full of lights). Could be something battery related as you've been told, but if that hasn't fixed it it may be your alternator or wiring preventing your vehicle from holding a charge.
If you have told a shop the problem and paid them money to fix and it has not been fixed then that would be the first place i would go to. Tell them what they did, did not help. Or if you do not want to go back ask friends and family for a shop they trust and get a second opinion!
Hope this helps!