You can marry, as long as you meet the state's eligibility requirements for a marriage license.
You cannot apply for a fiance visa while in the U.S.
If you have a two year home country residency requirement for your J1, you cannot adjust your status based upon marriage until you either fulfill it or obtain a waiver.
If you marry and don't seek permanent residency, the question of your intent to immigrate is likely going to arise if you seek an extension of your visa, or if you depart and reenter. If you do intend to seek permanent residency, AOS may not be compatible with your present visa and program; check with your program advisor.

