When I was shopping for an outboard I saw several "cheap" new outboards from several overseas "dealers". I contacted Yamaha about the "cheap" outboards and I was told that there are no authorized dealers and not able to buy or sale Yamaha outboards. The guy also said IF they actually sold...
Are you 100% positive the power switch on console is wired correctly and you didn't connect the hot 12v lead to the switched side? Are all accessories that are wired to the new fuse box on all of the time when you connect the battery cable?
Towing 400 miles is a big deal, you didn't mention if the route is all flat freeway or bumpy mountain roads or what you're towing with. I wouldn't do it without the proper trailer and tow vehicle but that's me.
I had the same problem with my lawnmower. It would only run while I'm pushing the primer bulb. The problem was the carb bowl gasket, causing the carb to suck air after running for a few seconds then dieing. Have you checked all of the gaskets? The problem doesn't sound like an issue with sand in...