Q: Car jumps in Reverse & less in Drive

For the past few months, when I put my car in Reverse, it jumps. Also in Drive, it slightly jumps but not as much as in Reverse. In Reverse, it is a much harder jump. In researching I’ve found that it could be anything from a simple transmission flush, to motor mounts, or a bad transmission. The electronic trip mileage acts strange, but then returns to normal functioning. Once the car has run a while, the engine light stays on. Then in a couple of days the light goes off and everything is fine until the next time, which sometimes can be weeks before happening again. It drives much the same as usual, except when accelerating from a stopped position it seems a bit delayed. The transmission fluid has never been checked because it’s encased and I can’t get to it. I’m hoping the problem is just a flush/fill needing to be performed. I had a friend take a look and he thought it was driving fine. When he started it, the light wasn’t on. Due to finances, I can’t take it to a dealership. Can you tell me what is the problem and is there any way I can fix this myself?