This takes me back, had to prove this theory mathematically during first year physics degree.
The longer you are out in the rain the more falls on your head.
The further you go in the rain the more rain you go through and ‘sweep’ out of the air.
So for fixed time in the rain you get wetter running as you cover more distance and hence ‘sweep’ up more rain with your frontal area. You’d stay driest just standing still.
For a fixed distance, eg to the LSB, you ‘sweep’ up the same amount rain whatever speed, but get less rain on your head if you run as you spend less time in the rain.
Furthermore as you will always ‘sweep’ up the same amount of frontal rain on your trip to the LBS and the top cross section is smaller than than frontal one trying to go quicker to keep drier starts to get in to dimishing returns, so only worthwile jogging not sprinting.
The above assumes constant rate of rain and deglects and increased spashing effect of running.