Did you really move to a new home, or did you just travel to the other state on a temporary assignment for the summer, while maintaining your permanent home in your original state of residence? Did you stay in the new state after the internship ended, or did you go back to your original home state?
What I'm getting at is that, depending on the facts and circumstances, this might be business travel expenses rather than moving expenses. Since you were an independent contractor, i.e. self-employed, business travel expenses could be deducted from your business income. Take a look at Chapter 1 of IRS Publication 463.
