Bike touring can be done inexpensively. There have been many savvy resourceful people who have toured for long periods on very little but you have to have some money. Even 2-3 months at a minimum wage job would give you enough to do a long tour. With this money you'd be able to get a hotel now and then when having a rough day, eat well out of grocery stores, treat yourself to a nice meal now and then, fix your bike if it breaks.... You want to be able to enjoy yourself not have to scrounge by on a daily basis.

You might learn something about yourself while on the road, it could be an unforgettable and wonderful trip, but to think that it's going to solve major challenges in your life is a mistake. I am not dismissing the idea of the trip just suggesting that it would be good to have reasonable expectations.

Best of luck with whatever you decide and here's wishing a great trip. Let us know how it goes or give us a link if you do a journal.