Apologies for the tone. Perhaps I should have taken a deep breath between finding out and venting on EF.

The thing that worries me is that now I am forced to sign a document that states that I will be leaving after the internship. But what if I receive another job offer to begin immediately after (which is the plan)? Will they deny me that opportunity based on this signed letter?

To me, it's kind of implicit that without a valid work permit, you need to leave unless you have another work permit processing. Why the need for this?