I'm really sorry to hear that. I have a rottie, who's the best friend I could possibly imagine. I can't imagine losing one let alone two.
Putting him down is the right thing to do. Your buddy depends on you to make kind, humane decisions for him...and at this point, putting him down sounds like the right thing to do for your friend.
Give him a special resting place, and if at all possible, see if you can have a vet come to your home to give him the shot. He'll be with family, calm and it's a very peaceful process.
My rottie Buddha has traveled all over the country with me. He's attacked the snowmen I build for him, swam in the Pacific chasing Canadian geese, chased bears out of my camp, been there for me every single time I needed a friend....every time. He's the only "person" who's happy every single time I come home, the only one sad every single time I leave. He can hold 3 tennis balls in his mouth at once and is more than happy to get up out of a deep sleep to "get the kitty" to stop doing some nonsense I don't feel like getting up for. He's more loyal, loving and impressive than the vast majority of people I have ever met....and he embodies qualities most humans will never achieve. He's just the best....I'm a lucky guy to have him.
My condolences to you.....gonna go hug Buddha now.