I am currently at University where I have met a girl who I like very much and would like to marry. However, I am not financially stable. Is there any situation where having an appropriate relationship before marriage is permissible in this situation?
Temporary marriage (Mut'ah marriage) is an Islamic marriage which saves those who can not afford permanent marriage. You and her can agree on Mahr and time for the contract. If she is a virgin Muslim girl, her father's permission will be required. She can put a condition on you not to take her virginity until you have permanent marriage.
You can read on this website more details about temporary marriage (Mut'ah marriage) or ask your local Shia mosque scholars about its rules in Shariah.
Wassalam.