I'm sorry you are going through this. I can remember taking one of my dogs in. He was old and in pain. He always hugged me when I cried by pushing his head on my chest. I was out in the car crying ready to take him in and he was hugging me. When I closed the door to leave him there he ran to the door. I thought I was going to die. I wasn't strong enough to stay with him when they put him to sleep. It is so hard.
My thoughts are with you and I'm sending big hugs your way. It gets easier with time. Their lives are just too short.