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TL;DR: Yes, no change for vertical areas but less wet in horizontal areas.
This makes intuitive sense to me just by looking at the two extremes, assuming a persistent and consistent everlasting rainstorm.
If you're moving at infinite speed the raindrops are effectively held in place so they don't fall on you; you only get wet from the drops you walk into on your way to your destination. Since the space between you and the destination is the same regardless of your speed you'll always get at least this wet.
If you've moving at 0 speed you never reach your destination so you're in the rain for ever and get infinitely wet.
I'm not sure what the argument for moving slower to stay drier is.
The argument is that running in the rain has extra hazards. Running across a now slippery surface is dangerous, increasing a risk of falling. Visibility is generally reduced for everyone (especially if they're wearing glasses), so your also at a higher risk of running into things or being hit by a vehicle. It's harder to detect and avoid puddles when you're moving faster. You'll also splash harder in those puddles, getting your shoes and pants wetter faster. Running through the rain really only makes any sense for VERY short trips or when you can see that you may be able to reach shelter before the rain gets significantly heavier. Otherwise, the difference in how wet you get isn't going be significant enough to risk slipping and falling, splashing through a puddle you might have been able to avoid at a walk, or some other incident because you were moving at a dangerous pace through a slippery and flooded world with reduced visibility.
Getting to your destination safely and unnecessary injury without soaking your socks and underwear is much more important than getting slightly less wet.