Do you really want to tour in places that have soup kitchens? That would seem to imply an urban setting. I know that would not be my cup of tea.

On the camping for free... On the Trans America we found it really easy to stay for free a good percentage of the time without the need for stealth. We found cheap campsites quite a bit too. On my Santa Fe Trail tour camping for free in town parks was pretty easy although I did splurge a few days on motels. In the Sierras camping for free was tougher. Some of the time we could have wild camped pretty easily, but many places we were on land where camping was specifically not allowed.

Personally I think going with no money sounds like a bad idea. Better to work for a bit before you go, sell some of your possessions, or raise money in some other manner before you go. You can go on a tight budget, but being completely broke will grow tiresome pretty quick especially in the wide open empty part of the country which you will have to pass through if doing a Trans America like the op says he (she?) wants to do. I think most people will be very surprised just how empty and desolate much of the country can be. When there isn't water for 80 miles on some sections, there probably aren't dumpsters to dive for thousand mile stretches in some places.