We had two dogs for 15 years who we had to put down, oddly, within a month of each other in Spring 2015. The sedative is a MUST. We told our kids (10 and 6 at the time) to say goodbye because we were taking the dog(s) to the vet and they may not be there when the kids got home from school. That might have been as much for the kids as for us, because neither my wife nor I had ever taken a dog to be put down before.
When it came time, I held the dog in my arms when the shots where given. I felt the life go from the body. Haunting, but I wouldn't want my dogs to die any other way. Peaceful in my arms.
As for getting another dog, you will know when (if) you are ready. We waited a bit and rescued a husky in April 2016 and then another in July 2017, so we again have two dogs. But it took a while.
EDIT: Might be cliche, but this gave me some comfort.
https://www.rainbowsbridge.com/Poem.htm