While I'm riding on a tour, I usually wear normal padded cycling shorts ... often under a pair of regular shorts.

If I wear underwear rather than my cycling shorts, I try to select a pair that is not 100% cotton. Cotton does indeed absorb the sweat ... and then it becomes very wet, and sits next to your skin all wet and salty ... very uncomfortable, and of course increasing your chance of rashes etc.

If wearing padded cycling shorts isn't your thing, try to find unpadded cycling shorts or something similar. Or at least partially synthetic underwear.