Quarantine was still required when we moved here the first time, so we chose to leave our four cats behind. Of all the moving issues, that was the most stressful. I cried and cried about it. Like Lindyloo, I begged my dad to take them. In the end, he caved and took two and a friend took our other two. If there is absolutely no trusted person you can find, I would search all over for a no-kill shelter. Ask other shelters, vets, animal groomers, ask everyone.
My parents had to give away our dog when they got divorced. Our vet knew someone who lived on a farm. We took him there and then went back a few times to visit and make sure he was happy. Although it's hard to leave your pet with a stranger, you have to trust your feelings. We got another of our dogs from a stranger who was too old to care for her and we gave her (the dog, not the stranger)

a loving home for the rest of her life.