I'm sorry about your breakup. It will be hard at times, and the anxiety is not fun to deal with, but yes, I think you can definitely do it, and you will. Have faith.
A 4.5 year relationship breakup isn't something to scoff at, and leaving your dog (dog lover, have 3:) has to be heartbreak on top of heartbreak. As the ex-wife of an airborne infantry soldier that deployed once to Iraq for a year, and once to Afghanistan for another year, I will say that our breakup and divorce was harder than the deployments. Both are frightening & painful, both hurt, both test your strength and resolve, but in two very different ways. I kept going during the deployments...I fell apart and dealt with a lot of anxiety and depression during my seperation and subsequent divorce.
You're entitled to feel your feelings and worry about how it may impact your education. Breakups are tough. I know. Hang in there. Hopefully, by the time classes start, you'll be a little more grounded and your heart will be on the mend. It takes time. It comes in waves. You will get through it and you will be stronger for it. School will help keep your mind focused on a different direction.
Best of luck to you. Chin up!