Just take two sets of cycling clothes, no more. Each evening when you take a shower, take the set of cycling clothes you wore that day and wash them in the shower as you take your shower. Wring them out and hang them up to dry. If they are not dry by bedtime, then just let them air dry the next day as you ride down the road. This only works if you are able to attach your remaining damp clothes to your bicycle as you ride.