I always run because I want to get out of the rain faster, but I wonder if I’m just running into more rain and actually ending up wetter…
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For this kind of question, it’s often useful to look at extreme cases.
Imagine you walked very very slowly, so that it took you many hours to go from one shelter to another. Even if the shelters were inches apart – so close that you wouldn’t normally get wet at all – the slow movement from one to the next would leave you out in the rain for ages. So very slow speeds get you very wet.
Now, imagine you moved ultra-fast, so fast that the rain is basically stationary from your point of view. You’d punch a “hole” through the rain shaped like you, like an old cartoon character plowing through a wall. That means you’d only get as wet as the amount of water directly in front of you, which is not very much. (And a slightly more subtle calculus argument will show that this is the minimum you could ever hit.)
While this isn’t a mathematical guarantee, problems of this kind tend to have one of two behaviors: either they ramp up or down to a maximum or minimum and then taper off, or they smoothly increase/decrease. In this case, it’s the latter, and you get the least wet by moving quickly. You strike the same amount of “forward water” regardless (assuming the rain rate is constant), but are exposed to less of the falling water.
That said, things like splashing in puddles, non-constant rain rates, different parts of your body being differently sheltered by an umbrella, etc can make this idealized answer not apply in practice.
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