It sounds like these folks are not able to be too cooperative, so I would suggest a trick I used with my kids when they were little. I would tell them that I was going to give them a beauty treatment or a facial. I would then take a warm (not too hot) washcloth and lay it over their eyes while I got their clothes, toothbrush, hairbrush, etc ready. Just a minute or two of warm water helped a lot in softening up the "gunk", making it much easier to wipe away. Also, even when they were very little, my girls hated me going near their eyes, so I taught them how to clean the gunk away using a clean spot on the washcloth. They were much more cooperative when they did it themselves. I don't know if your residents are capable of this, but it might be worth a try. You can hold a hand mirror for them.
Also, you may be correct that they do not have eye infections, but such copious amounts of drainage make me wonder about blocked tear ducts and/or allergies. So please ask the nurse to take a look, and ask whether massaging the ducts or using allergy drops may be helpful.