First - yes, drip lines work. The usual scheme is to tie a short length of shoe lace to the line. Cotton is better than the ubiquitous nylon or other synthetic. I'd imagine the cord/string to be about 7 inches long, You'd tie it at your desired break point with a garth hitch and have the two "tails" hang down about 2 to 3 inches.
But you mentioned "straps," not cord (like a whoopie sling). I have seen recommendations to put some twists in the webbing.
I haven't had a problem with my webbing because of the hardware between the webbing and the hammock. What do you have?
Tarps deserve a little mention. In rainy seasons, the tarp is hung below the ridgeline. If it were above the ridgeline, that ridgeline would create a path for water to run under the tarp. In windy conditions, that ridgeline would rub on the ridge seam, perhaps eroding any "water seal" treatment or just adding some abrasion to the seam. But in Winter, rain isn't an issue (around here), so I'd be more likely - depending on location - to run the ridgeline under the tarp for better support in a snow dump. I might also hang something on that line, adding a "buffer" between it and the tarp.
Back to the hammock - how are your straps connected to the hammock? Are you saying the rainwater is jumping that connection? How much of your straps are covered by your tarp - which is suspended below its ridgeline - right?
A photo of your strap/hammock connection would help a lot.