Stella, I am so sorry.
My old girl was pts a week ago for exactly the same reason. She was just shy of 13 and half and had started limping intermittently. We thought it was just a touch of stiffness/old age as it seemed to come and go and figured we'd get the vet to check her over when she had her vaccinations, as they were due shortly.
Then she suddenly stopped using her leg at all. Vet seemed pretty sure what it was likely to be and xrays confirmed it, the tumour was aggressive and had severely attacked a large proportion of her leg. They asked if we wanted them not to wake her from the anaesthetic, but I couldn't handle making the decision that quickly, so we brought her home and she was on anti-inflammatories and tramadol while we spent some time with her. She seemed a lot more comfortable, continued to eat well, enjoy cuddles, beg for treats and even ask us to play and throw her toys for her to retrieve, not to mention using her bad leg to scratch with.
She was checked weekly by the vets, the first appointment they were happy that the pain meds were keeping her comfortable, as she was happy to have it examined. Then within the next week we noticed she seemed to be coughing a bit. At the second appointment the vet said that she though she wasn't suffering too much and we arranged for her to have a chest xray later that week. When we got there they discovered her leg was unstable, although it hadn't been at the appointment two days before and the xray showed she had sustained a pathological fracture through the damaged bone - she hadn't shown any sign at all that the pain had increased, so we don't know when/how she did it. Xrays also showed numerous lung metastases. Of course we had to do the kindest thing at that point and say goodbye. 
The vet explained that some dogs are just incredibly stoic about pain and will keep on going regardless. It's apparently quite common with osteosarcoma. 
I am still wracked with guilt for not realising how much pain she must have been in before she reached the point where she stopped using her leg and of course feel even worse that I couldn't face letting her go before she ended up fracturing the leg. If I had been as brave as you, I would have let her go as soon as we found out what was wrong, but I just couldn't marry her still eating/playing and acting normally etc with her being ready to go. 
We had her with us for just under 3 weeks from initial diagnosis and what I do know is that during that time she felt safe and loved and that when she went she was ready and safe and comfortable in my arms.
You did the right thing by your boy. You did what you had to do, coming from a place of unconditional love and he couldn't ask any more that that of you.
Allow yourself all the time you need to grieve, but please don't blame yourself, you have nothing to feel guilty for.