Thanks everyone, for the thoughtful replies and good advice.
I bought some Charlie's Soap, and really like it.
However, while in Ireland I tried washing my synthetic shirts and still got the smell afterward. I did find out that it's very worthwhile to ask your hotel desk if they do bags of basic laundry for a single price. They didn't advertise it, but they did a pretty large bag of laundry for 15 Euros. This was midway through our trip and it gave us enough clean clothes to make it to the end.
I did try out a couple of Merino wool t-shirts. Unlike my synthetic, wicking, light shirts I bought at Costco for $6 each, the wool t-shirts cost $45 and $54, and that was on sale! They feel amazing and are incredibly light. They fit well, and after hand washing, dried quickly with no smell. I think I need some inflatable hangers, though, since there is a slight 'bump' on the shoulders from drying on my normal hangers. Taking cost into consideration, I think I will travel with a bunch of the inexpensive, light, wicking synthetics, and two wool shirts. If I can't find a place to do my laundry I will live with my two wool shirts until I do find a place. The t-shirts were WoolX and IceBreaker brand.
In Ireland I wore a new pair of Bluffworks chinos. I really like the fit, look and feel of them. They look like normal pants, not technical travel pants. They have anti-pickpocket features, and the chinos have a very soft feel to them, rather than the synthetic feel of their original pants. I could wear these for a week if needed.
And thanks for the excuse to wear natural fibers on the plane. I now have an actual reason to wear my jeans, not that I needed on.
Happy travels!