What company owns your Starbucks?
Find out who the person above your manager is and write a proper email to them, include the HR email as well and make it clear that your availability was spoken of and that this is written record to them of what your availability is.

Your manager is clearly not going to listen so that's where the above management comes in. HR will ensure there should be zero conflict between parties during this as it's their responsibility to ensure the business is compliant with all laws and protects their interests.

And if that fails, quit without notice. If they can't get you what you want then you simply do not work for them any longer, trust me college students are in DESPERATE supply so you'll find work anywhere