A decent chain lube should be able to take some wet weather in stride. If it washed off, you can use some more, or oil scavenged at a gas station. IMO the goal of a bike tour is to enjoy the enjoy the bike tour.

Assign a carefree tour a higher priority than worrying about your drivetrain. Decent lubes will last hundreds (or more) miles , so feel free to ignore your chain a bit. By now you should know the relube frequency for whatever you use, so just stick with that. As for chain stretch, you should be able to get across the country before it begins to be an issue, and if you do nothing at all, you can probably ride back again without problems, though it might cost you the cassette.

For my part, I'll gladly sacrifice some drivetain hardware to enjoy my trip more, but to each his own on that score. Ride NYC to San Diego, buy chain (if you want) and ride back.