If you're planning to do lots of riding in the future, you'll need a new cassette soon enough anyway. You might as well buy it now, to go with your new chain and give it a try. Cassette tools are cheap enough and easy to use, and you can get by without a chain whip (if you want to save dough) by wrapping a rag or old belt around the cassette and holding it with a pair of pump pliers (Channelloks, by any other name).

If the new cassette solves the problem, that's it. If not, switch back, and save the new one for the future, and look for another cause, which could be one of many, but I started with chain wear because the problem started with the change.