How do I stay in the US if I am a nonimmigrant marrying a permanent resident?

Significantly more information is needed in order to provide a legal analysis of ways you may be able to attain your goals. That may include, for example, information about your current nonimmigrant visa, your immigration-related history, the date your fiance became a Permanent Resident, etc. There really is no substitute for you and your fiance to consult with an immigration attorney who, after learning all of the relevant information, could advise about eligibilities, options and strategies to attain your goals, and who then could offer legal representation in the often complex application process.