Based on all of the information you have provided, I would definitely choose to move if I were in your shoes. It sounds like the quality of life will be significantly higher, and the incremental cost will not be all that high once accounting for the lower gas costs and savings from cutting cable. If I did choose to stay with my parents after I had acquired a full-time job, I wouldn't feel comfortable living in their house without paying at least a modest rent, and that would further close the gap in costs.
Even after your move, it looks like you could have a 30%+ savings rate off the bat and a much higher savings rate after finishing off the car loan. That sounds like a great start to me and will put you in a very good long-term position as long as you can focus on maintaining your lifestyle as you progress in your career.
Assuming that you do move, I think it would makes a lot of sense to try to find a place where you are able to either ditch the car entirely (which could be a huge benefit to your finances) or at least avoid having to use it daily for your commute. It's possible that selling the car might not be feasible if you frequently drive back to visit your parents, for example. But even so, quickly running the numbers for owning vs. renting a car when one is needed could provide more information to make a decision.