Yeah, that's graduation week, so you aren't going to find much for cheap accommodations, pretty much anywhere around. And lots of people, too. I would probably choose something more suburban for a hotel, if you are looking at north of Boston try the Doubletree in Bedford Glenn. Not the greatest hotel, but certainly decent enough. Sounds like you will have a car, so from there you can drive to Alewife station on the Red Line and get into Boston that way. You can try the Air B&B but I bet they wont have many better options.
Keep in ind for your plans - lots of New England is based around the smaller towns, not just the big cities. Certainly stop in Providence and Portsmouth, but also maybe look at Lexington, MA and Ogunquit, Maine. Alternatively, spend a few days on the cape, not just P town but Hyannis and Sandwich.