I usually walk to the laundromat to pick up clothes I've left there ironing, but instead of carrying all the clothes with their hangers squeezing my hands out of circulation (it's actually painful when I carry them on the hand that has the wedding ring) I game up with a simple solution. I grabbed a garbage bag, twirled it around like a rope, made loops on each end, and then I stick the hooks of half the load on one of the loops and the other half on the other loop. Then I carry the whole load on my shoulder, with the clothes in front and back, balancing each other.
Hell, I've done it with my belt even when picking up groceries. Just take it off, loop it around the bag holders, buckle it, carry half of the load on the back and the other half on the front, leaving my hands free for whatever, be it hailing a cab, texting, carrying an umbrella or what have you, since the body is doing the lifting.
If someone invented a small strap, with a cushioned side for the shoulder and metal hooks on each end, I'd buy it. I figure this has to exist already but I've never seen it being sold.