Is it permissible for me to use the money I won in a competition to pay off a university debt that is preventing me from graduating, and then later save another amount to start the project for which I won?

It is impermissible to spend the grant on anything other than what the granting entity specified, because a Muslim is commanded to fulfill covenants and promises, as stated in His — Exalted be He — sayings: (O YOU WHO HAVE BELIEVED, FULFILL [ALL] CONTRACTS) and (AND FULFILL [EVERY] COVENANT. INDEED, EVERY COVENANT WILL BE QUESTIONED). Therefore, the principle is that what has been taken for a specific purpose may not be spent on anything else except with the permission of the grantor.