When your U.S. citizen spouse petitions you for permanent residence (i.e., green card), you can apply for an Employment Authorization Document (i.e., work permit) at the same time. The work permit will enable you to find employment outside of your school. The work permit is usually issued within 3 months of applying. After obtaining your work permit, you may apply for a social security number, if you do not already have one.
You do not give up your British citizenship when you apply for green card. Green Card simply authorizes foreign citizens to reside in the U.S. on a long-term basis. You are not a U.S. citizen when you obtain a green card. You continue to be a British citizen living in the U.S.
Regarding health insurance, once you are married, your spouse should be able to add you to his health insurance right away. Some health insurance companies require social security number (SSN) in order to add the spouse to the policy. If you do not have an SSN, then you can apply for it after you obtain a work permit. Once you have an SSN, you can be added to your spouses' health insurance.
As for school tuition, this is a question for your school. You should consult your school on how to qualify for reduced tuition, if any.
For more information regarding green card process, visit this link:
http://bccvisalaw.com/adjustment-of-status.php
For more information regarding the work permit, visit this link:
http://bccvisalaw.com/work-permit-ead-card.php