You can tour on 'bout any bicycle that's mechanically sound. A mtb will do. What's most important is that it's a good fit and has a granny of at least 20 gear inches.

Free camping, $25/day will get you cross country in reasonable comfort with no weight loss. More is better. Subtract what you normally spend when home to get net cost for the tour.

One set of good tires should survive a x-country. Always have two spare tubes. Most that can go wrong will fail slowly and can be fixed at the next lbs, or roadside depending on your skills/tools.

Train or bus is often the cheapest way to ship a bicycle. Depending on the airline, it may be the cheapest. Box the bike.