I would definitely try to come to earlier orientation. If you don't grad until June 13, does that mean you have some exam days before school actually ends that you can skip (if you exempt exams). The biggest problem is for people with specific major requirements - such as nursing or engineering. If this is not you and you just need 5 classes to start, you may be okay, just be prepared for bad schedule!
I am hoping to go during the first week. That way its not too close to the 1 final I'm probably going to have to take, and all the graduation festivities! Thank you!