Sounds like you are hesitant to stay there as well. As a designer you need to speak up…it’s not a good practice to work on assumptions or fear of what could happen.
It’s just business so just ask the right person when you can chat about full time employment. Be grateful they hired you as an intern but don’t feel obligated to stay…specially if you have other better jobs lined up.
Whatever you do, just don’t jerk them around and don’t let them jerk you around either. If you are looking, be upfront and let them know if necessary. If you have offers then find out theirs and then make a decision. You can negotiate but if you are forcing the salary or working conditions then it’s probably best if you take the other job.
Good luck.