You could get a hostel somewhere for the 5 weeks and busk when you get a chance to help pay for your room. Doing a quick search the cheapest I found was $20/bed/night, but I'm sure you could get cheaper, especially if you asked around and let people know you were looking for a longer than average stay.
On a side note, if you need to get somewhere that's too far to walk, I'd advise the subway - dead easy to navigate and I think it was $7.50 for the whole day, or $2 for a single. I know you guys are on a super-tight budget, but that is the cheapest way of getting around outside of walking.
Wish I was doing something like that - went last year, but only for 5 days!
Close - acoustically