Hi, welcome to the forum.
Try going around and looking at chicken bums. Sometimes you can spot the chicken having a problem. Our Hermione laid two quite bloody eggs last spring, and I could see a correspondingly bloody area just inside her vent. We're not really sure what caused it, but we encouraged her to go broody so she would get a break from laying and heal up. After she started laying again, she was fine. Our vet did recommend that we euthanize her if she continued to bleed significantly with laying, because that would mean she was tearing internally.
If it's just a streak or two of blood, that's fairly common, though, especially with young hens as they're starting out laying. And as far as the stickiness or wetness, is it possible that you happened upon a freshly laid egg? Every egg is moist with bloom right after it is laid. As the egg cools it dries.
Post a picture of one of the bloody eggs so we can tell what you're looking at.