Building a Multi Hull
I figured that the draft issue might be partially what is driving your decision toward a multihull but I had not even remotely considered that the possibility of sailing the boat down the Wabash River to the Gulf of Mexico as an element shaping your decision.
While I have no idea what the draft of the Wabash is, I would suggest that the key element of your goal to sail around the world is toughness and seaworthiness. While there certainly are catamarans that can do the Missippi part, and there might be cats that can do the circumnavigation part, and there are cats that you can build yourself pretty cheaply, I would suggest that for the kind of dollars that you are talking about, even if only for materials (maybe , it is unlikely that you can find a cat that can do all three.
In the big picture, given the sheer amount of money that it would cost to build a boat like even a small Wharram cat (something approaching $40 K I would guess), you would be far better off either buying a used monohull that is of a suitable design and spending a little time and money putting it in shape or else building a more suitable mono-hull even if it means trucking it to the Missippi and then commissioning it there.
Even if you decide to do a Wharram, the likelihood is that you will be predominantly motoring down the Wabash and Missippi. Wharrams are not very good boats to sail in close quarters being unreliable to tack and manuever. If you really want to do a multi-hull I think that you would do better with a small Trimarran rather than a cat. I would note that any multihull capable of going offshore with a couple on board is unlikely to be suitable for pulling up on a beach.