I used to use the rubber disks that are given away to help open jars, but I have a much better technique now. We found we still made a mess washing in the sink so I changed to something that washes better and keeps the mess contained.
I use a zip lock bag. I think I've said before it is a gallon zip lock, I was mistaken. It is at least two gallons and I think it's called a big bag from Hefty or Zip-Lock.
My daily wash is a pair of socks and underwear with maybe a t-shirt added (all non-cotton of course). I put the clothes in the bag, add a very small squirt of concentrated detergent and fill the bag about half-way with warm water. Seal it, shake well, open the corner to drain into the sink, then add rinse water, shake, drain and repeat until no more soap suds are evident in the rinse water.
For drying, I use a car chamois to wring them out before hanging on my twisty nylon line. I've found the chamois to be a million times better than using a room towel or travel towel. It pulls more water out of the clothing and then wrings to almost completely dry for wringing more clothes or line drying. A 5.5 square foot chamois can be had at Costco for about $15.
If you're looking for travel towels. Costco also has a set of three car wash mits and two towels that are made from the same material as travel towels. The towels are much cheaper and better than the travel towels we already had in our travel gear.