Yebbut firstly hotels and some hostels give/rent you towels so you don't have to carry that water away with you and secondly even if you use a travel towel, you've now got two items roughly half as wet so there's a chance that they'll both dry before morning. If anything, travel towels seem to dry more quickly than most clothes.
If I was doing lejog over a 2/3 week period I'd book in to various B & Bs enroute & post out clean clothes to them before I set off & post all my laundry home 
that way you only need to carry 2-3 days worth of clothes 
In fact on my first ever solo tour (C2C) I'd taken clean everything for each day, after day 2, I mailed most of my stuff home, made the climbs after that so much easier 
The mailing trick is something I intend to try, but probably mail to myself at post offices as I tend to camp, and sometimes arrive late / leave early. If I use post offices, I can send to ones I expect to pass during business hours.Think this rinsing cycling clothes out depends just how sweaty you get. I run really hot & can get sweaty even in winter & I've tried rinsing tops out in the shower, but they still stink the next day
