My questions is are they allowed to send me to collections without ever trying to notify me that I owe money.
Yep. But that's not what happened here. They DID try to notify you, but you neglected to update your contact information when you left school. It's not their obligation to try to track you down.

You can try to negotiate with the CA about the late fees, but they are under no obligation to take a penny less than what is now owing. As Disagreeable notes, all you had to do was pay attention to your tuition statements to know that something was amiss. The University is not at fault here.