We all go through the learning process one way or the other. Filling tanks because I didn't plan ahead is one I'd rather not repeat too often. In fact, at some campgrounds that don't have sewer at each site, you may also want to dump "before" you set up. At a state park in Colorado we had to disconnect in evening to dump, then reconnect. This happened because I didn't dump before setting up thinking it was just one night and I'd do it in morning on way out. That was a mistake.
To your question, yes -- dump black first. Then grey so it rinses your hose as much as possible. Then after you disconnect hose, use their non-potable water supply at dump station to rinse hose as clean as you want/need. It helps greatly to have disposable gloves (consider it a requirement), so buy a box before starting out.
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