Need some help with 2001 Dodge Neon....

Hi, I have a 2001 Neon and I am having some issues with it. First of all, sometimes, when I put my car in reverse and am turning the wheel, it sputters like it is fixing to die. As soon as I straighten out the wheel a little, it stops. Everytime, I go to take off, it sputters like its not getting enough gas or it has sucked up some dirt. It is getting worse. Also, while I am driving, you can feel it sputtering. My husband can't seem to figure out what is doing it. I don't know if my car has any kind of fuel filter or anything, we change the oil like clockwork. It's just kind of scary when my 3 girls are in the car, and I go to pull out of a parking lot or cross a street and the car acts like its going to kill on me something and get me killed. A few months ago, we had a problem with the gas getting stuck on it. The casing around some wire or something had melted or shrunk, so whenever I pressed the gas, it would catch and stay for a while before it would release. For a week, all we had to do was crank it, and it would catch and idle really high. Could that be causing the problem I have now, or is there something I need to check. Also, the air filter has been changed. Thank you for taking the time. Anyone who might have any ideas about it, please let me know.