Don't post here often but I had a similar experience about a month ago. My roommate was out of town and his dog got loose by busting through our fence when I let him out to go to the bathroom. Naturally, I lost my shit and began searching all over. After about an hour we found him on the humane society site. What I wish I could have told myself back then was "don't freak out!"
The experience with the humane society here was (all things considered) pleasant. They were patient with me, and after a few phone calls and a minor amount of money, dog was out and back home... wagging tail and tongue out like he had the time of his life.
General rule of thumb when I go anywhere like that is to bring everything you have and let them decide what they do and don't need. I went in with my ID and proof of his rabies vaccination and got him out fine after they talked to my roommate on the phone but the more the better.
Can't speak to wait times. Mine was about 2 hours, but that was at 1pm on a Saturday.
Best of luck. Keep us updated.