I'm pretty sure it's rinse them warm; Cold would cause them to contract and pull in bacteria.
I've also heard that you should gently sand them with sandpaper if they're dirty.
Since yours are sticky with egg, I'd be inclined to give them a quick rinse and let them air-dry on a towel before putting them in the incubator.
And then I'd watch them closely to make sure they don't seep too badly after you put them in the incubator.
The protective coating is probably already compromised by the egg goop.
And honestly, after that much trauma and temp issues, you probably won't get a hatch anyway. Sorry!!