Rescue agencies can be really critical. They tend not to see kids as being a good enough reason to rehome. I’d use them as a last resort.
I’d start with seeing if you can put up a note at your vets. They might have people who have other pets & they can vouch for both you and the other person.
We rehomed a dog who hated one of my four kids on Kajiji. We met the people, saw where they lived and vetted them in person. It worked out well. He went to live on a literal farm with teenagers. Much better for him.
Rehoming is hard.