Well, we need some help....how about some pictures.
You say you "think" you got the right part. That is a risky assumption in plumbing. We would like to see pictures of the old and new one. If this is a typical 2 or 3 handle shower, then yes the seat inside needs to be replaced. Yes, you need to buy another $8 tool to do that, or you could pay a plumber $100 for a service call. He will have the tools!
There is probably a gland nut which tightens up the packing around the stem. The stem should be well lubricated, then tightening that by the proper amount will allow smooth operation with no leaks.