Yes, wash them every night. I usually take my shorts into the shower with me (if I have a shower available). You don't need to necessarily do the world's best job of washing, but make sure you thoroughly rinse them out (in as clean a water as you can find--it doesn't do a lot of good to rinse them in polluted water). Wring them out well. Roll them in your microfiber towel a few times and wring out the towel between rollings. That will get them quite a bit dry immediately.

In dry climates, they should be dry by morning. In damp climates, maybe not. That's why you should always have a second pair. Two pair should be enough most of the time.