Angela-
Your words are wise. To test this, I called the agency just now to have a conversation and feel things out. I think their mission is to move animals. This is understandable, but this means I need to have my radar up bigtime. Its going to require more judgement than I had anticipated. Glad to hear you are happy in spite of your pup's health conditions.
The name of the agency here is "Friends of Greyhounds".
http://www.friendsofgreyhounds.org/FoGSoFla.html