This is one where I KNOW I do it wrong. The dentist prescribed me a hardcore sensitivity toothpaste for an issue, but emphasized that it will only work if I don't rinse after brushing. But... I just can't not rinse after 30 years of muscle memory. I generally remember this right as I'm rinsing. The sensitivity toothpaste has not worked, of course.
I’m in the same boat as you — sensitive teeth, but can’t not rinse. What works for me is to treat the toothpaste as a “post-rinse medicated rub”: squeeze out a bit more, and rub it over the most sensitive teeth.
If you remember it as you're rinsing, brush your teeth lightly again to spread the tooth paste on the teeth. Once you've had yo do it again a few times, you'll pick up the habit of not rinsing.
What I find helps is to rinse my tongue (run the bristles over my tongue under water, but just the tongue). Then at least most of the toothpaste residue sticks to my mouth and teeth but not the taste.
Maybe mark the cup with something that will remind you. Clip a paperclip over the edge? Store it upside down? (If you're using a cup.) This could help interrupt the habit.
