Perhaps go with a more earthy colourscheme, I would look into organic food packaging designs for some colour theme inspiration if I was you. On the basis that they look after farmyard animals, I'm not sure the paw idea will work or be relevant..
I also though it was meant to be a paw... but the turkey idea could work, (I think quite unusual which is good)! as now when I look at it I can see a turkey as if you were looking at it straight on and maybe you could put a beak on it using a simple triangle pointing to the right. And maybe play round with the feathers try fading them out to look more feather like!
I dont see a turkey even looking again now. To get to the point as I am a bit tired, why a turkey? Is it a turkey sanctuary? If yes, fine. If no, why? I think most people will see a paw first as it says animal sanctuary.
Its a nice idea, but I'm not sold. I have produced some gibberish/nonesense logos in my time, so I can't really say much, but in this case, it is nagging me.
I would go in the direction of a shelter / farmhouse logo similar to your second logo, because its not just for one particular farm animal, yet all farm animals do live on a farm. i would go for something that represents a farm, a lot more care and attention to this icon though, because i think its key to have this looking good, after all logo text cant be that funky but the icon can.