So sorry you have this hard thing to deal with... we have had to do this a few times for various reasons and it is good for you to have a plan to fall back on no matter whether you can find him a home or not. You never know when a bird is going to be injured in such a way that the kind thing to do is put them down etc. My DH has done this and tried a number of ways and what he prefers is the quick blow with an axe or some other sharp blade. The thing is to do it at night if possible (they are much calmer and easy to catch). DH has a chunk of wood that he lays them on and then one hard blow and it is all over, they will still thrash around, but at least you know they are not experiencing pain and fear as their head is off. Warning though, the axe needs to be sharp and you need to be sure to hit hard enough... we had a couple of incidents early on while we were learning all this stuff that was just awful because of not wanting to do it, we held back and injured the bird instead!!!!!!!!!!!!! NOT nice and definitely traumatic... bless you as you walk through this. Hopefully you find a home for him, but remember to have a plan B for one of those horrid circumstances that will need you to take action for their good too.