For most, the extra line that hangs off the sling is sufficient enough to divert the water. But in an area like yours, you may benefit from keeping the suspension separate from the hammock. Like you said, you really only need enough strap to wrap around the tree. So if you attach the fixed eye of the whoopie sling to an end of the tree hugger, and use some metal hardware to attach the adjustable end to the hammock like a carabiner or Dutch's whoopie hook
which is a tiny piece of titanium. That way, you have some metal hardware to break the water's path, and you can keep you soaking wet lines packed separate from your hammock.
Also, check out The Ultimate Hang.
Great book, and website as some great illustrations and more.