2000 Sentra Electrical/Transmission Problem

My Sentra is a 1.8L with an automatic transmission and a few days ago I accidentally left my headlights on and my battery died, which has happened a few other times through out the lifetime this paticular battery. The battery itself isn't extremely old, but this particular problem has happend before right after having my battery jumped off. Right after having it jumped off I would give my battery enough time to recharge itself then I would go to leave. When I shifted into reverse the car ran smoothly, but when I shifted it into drive and went to accelerate my car was extremely sluggish like It was stuck in a higher gear. So I stopped and shut the car off and turned it back on and it ran perfectly from then on out. This has only happend maybe twice, right after I had to have my car jumpped off. But now the battery had to be jumped off again recently and I am having this problem alot. Now I might have to turn my car on and off several times before it will shift right and if I leave it running after it does work correctly it will shift properly even if I put it in park or reverse or whatever... until I cut it back off then I have to recrank it several times. It also seems to have aliittle trouble cranking. Could a new battery solve the problem?