Could I get approved for a car loan?

Not likely with that score. If you do get approved you'll have an ugly interest rate. What do you mean, your score keeps going down because of your student loans? If you're not making the required payments in a timely fashion, I would highly suggest taking that $5000, buying a used car cash, and starting to repay your debts as you agreed. Student loans that are deferred or have proper payments being made only do good things to a score. Good luck!