Most cotton dress shirts are not dry cleaned. They go through a laundering process: washed in water, finished with starch if requested, and pressed by hand or on a shirt press. Water cleans cotton better than solvent does, and the press gives the collar and cuffs the crisp finish.
Linen, silk, or specialty blend shirts often go to dry cleaning instead, because water can shrink linen or distort silk weave.
At Joy you can request your preferred starch level (none, light, medium, or heavy) and folded or on a hanger.