Sewing machine problem: machine won't tighten the loops under the material
The bobbin case rocking back and forth is normal! In order to form a stitch, the top thread has to cross under the bobbin thread and back up. It does that by looping all the way around the bobbin case - you're seeing the wiggle room to allow that, called the escapement.

It is hard to say what the problem might be from your question, but has the machine been oiled recently? Cleaning, oiling, and new needle cure a lot of machines.