I haven't made a habit of sink washing a shirt I would wear with a tie. (Although I have done this in a pinch).

I generally will pack 2 or 3 dress shirts and have them laundered by the hotel.

For days that do not require a tie I pack some sort of travel shirt and wash in the sink. Key thing to avoid is knit cotton which takes a long time to dry. Columbia has some decent shirts, some of which are 100% cotton and dry nicely overnight. You can look at other brands such as North Face and Royal Robbins also. Basically look for high percentage of synthetic fabric or a seersucker-type cotton that will hide wrinkles. Plenty of options with no pineapple or Haiwaiin dancers, perfectly fine for business casual.