I agree budget is likely to be a key factor. Given your interest in starting not too far from Memphis, I did a quick search of "Natchez Trace Bicycle Tours" since the Natchez Trace is a reasonable ride for a tour and not too far from Memphis. I can find a number of companies doing such tours on the Trace:
1. This one does pretty easy days, van support and staying in inns: http://www.vbt.com/tours/mississippi
2. This one rides the trace in about seven days: http://www.blackbearadventures.com/NatchezTrace.html
3. This one is in southern parts with moderate mileage: http://escapadesbiketours.com/our-to...race-bike-tour

All of them stay in lodging, have van support. You're paying enough that they should be able to talk in advance on accommodating particular needs. However they are also pretty expensive. Given those costs and the relative ease one can do a ride on the Natchez Trace, I'd be tempted to instead put together my own tour. Perhaps get some ideas from a book such as http://www.amazon.com/Bicycling-Natc.../dp/0963779869 and put together your own tour with some friends.

A different and less costly alternative but might take some more work would be to take an event type ride such as BRAG (http://www.brag.org/index.html) that is normally set up for camping and group meals and instead see if you can book motels and get meals along the way. For a ride I am doing at end of June (Bicycle Tour of Colorado), there is a company that offers hotel packages: http://bicycletourcolorado.com/index...display&pid=31 and perhaps there is something similar on other event rides closer to you. Using BTC as an example, meals you would likely need to get from communities along the way.

In summary, my guess is if you are willing to pay for an expensive tour, there are companies I expect should be able to work with you. If not, both the Natchez Trace and Katy Trail would be my candidates for putting something together myself. In addition might look at adapting an event type ride such as BRAG or BRAT (http://thebrat.org/) but then see what you can do about indoor lodging and food.