Staying Down
Many reasons for moving on the shot or jumping up, but the main reason I see people do it is they are wanting to see the result of the shot.
Best Quick Fix for this - Gently press bridge hand fingers against the table. This will anchor you to the table. It will no longer be natural to just jump up, instead you would have to push up off the table and its not going to happen. Try it, and you don't have to press hard, just enough that you feel the table pushing back.
Also the biggest jump I ever had in my game wasn't what I expected. It was a different mindset. What do we have control over? We have control over what we think about before, during, after the shot. We have control over decision of shot, how we setup for the shot. We have control over our stroke. We do not have control over the cue ball, the object ball or the outcome of the shot. If you can wrap your mind around that and focus on what we have control over - its eye opening. On my final stroke I am no longer thinking of cue ball, aim, position, spin, speed, pocket... all I think about on the final stroke is keep my head perfectly still and deliver the best stroke I can (that is the same as what I did during practice strokes).