Yes, I have felt like that, and YES YOU DID THE RIGHT THING!
I lost my first ever dog with bone cancer, she was a GSD and was only 6 years old. It is nearly 20 years ago now and I still feel the pain of parting with her. She was a rescue I took on at a year old and she had sever HD.
I have since read about people going for treatment and amputation, but even today dogs only survive approx. 1 year.
This is a terribly painful condition, I had my girl put to sleep a week after diagnosis, so you are lucky to have had so long with him. At least he was in his own home and had familiar scents around him. They do just go to sleep, my girl had gone before the injection was even over.
I recently spoke to someone who had just lost a dog with this in the front leg, they had actually just left the vets and the dog stumbled, when it got up it could not put the paw on the ground, they went straight back in the vets had the dog x-rayed and found out the bone had disintegrated. The dog did not come out of the anaesthetic. The only treatment for this condition is amputation of the limb and follow up chemo. etc.
That is what would have happened to your dog if you had not let him go. I thank god that I let my girl go before she broke her leg.
I too felt initial relief, that is a stage of grief, just like the guilt you are feeling now. Eventually you will be able to remember the good times and some of the pain will fade.
To this day I do not regret my decision to let her go.
Take care.