I have just been through exactly the same thing unfortunately. My DP had a GSD when I met him, a very large, active, bouncy, lovely dog. He had pretty good health all his life, then a few months ago he started to dribble wee all the time (he was 12). The vet gave him antibiotics which didn't do anything to help. Then over a few more weeks, the dribbling turned into full on incontinence. DP was in total denial. I kept saying 'it's not fair on the dog, he is lying in his own wee all the time, he's trying to lick it up, he just looks so miserable', but DP didn't want to get the vet out again because 'he was alright in himself' and apparently the only reason I was saying anything was because 'I was sick of cleaning up after him all the time'. Of course, I was sick of it, but I knew the dog wasn't happy!
A few more weeks went by, and things obviously didn't improve, so we got the vet out again, who gave him more anti-b's, but said that if they didn't work, it was more than likely a tumour, or damage to his nervous system, in which case it would be very expensive treatment which may or may not work.
Fast forward to last weekend, he went downhill very fast. His back legs went and he was struggling so much, it was heartbreaking to see. At that point DP did the right thing. It was just so sad.
After an emergency vet botched putting him to sleep on the Sunday (that's a whole different thread), we got our regular vet out on Monday who assisted him to go. It was very upsetting for us, but he just slipped away.
I think if your DP is anything like mine, he will just realize, hopefully sooner rather than later that it is the kindest thing to do. DP wanted to hang onto the old boy as long as he could, and in a way, I did too, but it wasn't fair on the dog.
Big unmumnetty hugs to you both, its such a hard decision to make x