Up until recently I hadn't experienced this.
However with my recent purchase of some Corrine Dennis shorts I found that the fit around the genitals was a bit more generous, and that although everything was fine at the start, after a stop (rest, traffic lights etc.) my penis would drop to one side and shortly after that would start to chafe.
Rooting around down the front of your shorts is not a good look, and also quite difficult whilst trying to maintain a good pedalling rhythm.
I did find, much to my surprise, that wearing underpants sorted this without adding to the riding discomfort. Then again these are very skinny Rohan ones. I wouldn't like to try it with M&S cotton ones.
Up until now I have used the more traditional Lycra shorts which have a much more "grippy" fit which has kept everything where it should be.
For protection against chafing I have always used baby lotion (recommended when I bought my first cycling shorts as an effective budget alternative to chamois cream). This works well for me.