I second the idea that you need to eliminate one thing at a time. Unfortunately with rear dérailleur issues, the first problem to eliminate should always be to check for a bent hanger if you can. Especially if you tried adjusting tension and limit screws and still have problems.
Deciding whether you want to buy the appropriate Park tool will depend on how many times you think you'd use it. I've already straightened 5 bent hangers in the last 18 months. But then again there are a lot of bikes in the family and I also do this for a few friends.