Is it possible that this problem might be assumed and not factual? Meaning that you assume the tampon gets soaked through when you pee, but in reality it’s not? I’m asking as that has never happened to me and as others have already pointed out, the tampon is quite deep in. The string doesn’t have a huge suction power where it basically starts soaking more and more pee inside you via the string. In my experience all that happens is it might get a bit wet every now and then but that’s it. Then you just dry it with toilet paper. My string never hangs out, it’s always tugged away, also when peeing. But I guess it depends on anatomy how well the string stays tugged away.
Maybe pee with a tampon and when you think again that it gets all soaked in pee take it out and have a look. Then when you see there is no pee, hopefully it will help clear your mind of the misconception it made and you won’t have this problem any further.