Well, I'm a science instructor, and science is by nature a shared learning type of enterprise. My students do labs and research work in groups and other work individually. One thing I have noticed, and that you may want to think about, is that the groups tend to function much better if there is one "take-charge" person in the group. Those who have good leadership skills usually function as a project manager, setting the tone right away by going over the task with the rest of the group and making sure everyone grasps the mission and the time line, breaking the task into sections, and then clearly asking each member what sections he or she is going to take responsibility for. (This works much better than assigning tasks, psychologically.) The leader then does periodic check-ins or redirects other group members. Yes, the leader is doing more work than the rest, but is doing considerably less work than if he or she were doing the whole job alone, as seems to be happening to you.
Before you say you can't do that, consider this: leadership skills are learned, and are an incredibly valuable commodity in the real world. If you come out of college having learned only leadership skills and nothing else, college will have been worthwhile and will pay off in the job market. On the other hand, without such skills you can be the smartest and most knowledgeable person in the whole company, and still never advance to a better position. You do not have to be a "social person" to be a leader; sometimes, the best leaders are actually somewhat antisocial, because they have a thicker skin and don't really care if the others in the group think they are being pushy or whatever; they are not there to make friends, only to get the job done.
From the tone of your question, you sound like a responsible person who cares about doing well. Although it may go against your grain to function as a project manager, I would suggest you try it. You can't change the assignments or the grouping, but you can change your approach to the work.
Here are a few links to articles about project management and the basic skill set and approach:
Six essential project management skills
Seven key skills of a project manager
Ten skills for project management success