I know there's various feelings on this forum that span the range between pets vs livestock, but that rooster (or hen) should probably be culled instead of re-homed. In a large farm free-range environment there'd be too big a risk of passing on faulty genetics (IMO). In a backyard situation...
So, not sure if this is relevant but here goes - I had a cat that did that. Her whole body would micro-vibrate almost all the time. She was the runt, and that's just what she did. And she purred almost all the time as well, it was like a safety valve for her, she'd even purr in between...