You are NOT a bad mother, upsetting as it will undoubtably be it will actually be good for them to see you grieve. Kids need to see emotions and see that they are normal.
My post looks really harsh now I read it back and it may be that I am projecting, but unfortunately it's what i did. 
My favourite ever cat (I know you shouldn't have favourites but I did so there!)started messing in random places having never ever gone in the house before, not even in a litter tray. I knew what it meant but just told myself it was "age" and he was clearly fine. He slowly started losing weight, again just "age", it happens to all of them.
I would talk to my friends about it all the time, kept re-itterating how normal it all was and how healthy he was in every other way.
Got him a litter tray and all was fine for a while. Before I knew it I was having to completely empty and re-fill it every day. He got really fussy with food and would only eat 1 specific flavour of 1 specific brand. All just normal "aging".
One friday he came in and wouldn't lie down properly, just kind of hunched down and wouldn't properly settle. Told everyone I would take him in on monday if he was no better. He was still purring so couldn't be that bad.
Long story short vet did tests and found out he was seriously ill with several things and had been for a long time, 2 years possibly. He was suffering badly. Vet said I could take him home and bring him back "when the time was right".
I so very nearly did but at the last minute, thankfully, I crumbled and told him to do it there and then. I knew it was time, I'd known it for a long time just couldn't accept it.
If you're really honest with yourself is your vet really the only one who knows best, or is he the only one who tells you what you want to hear, is it just an excuse to wait another couple of weeks? Every vet I've ever known, regardless of whether I like them or not really does want what's best for the animal. (The only ones who don't are the, thankfully, very rare ones that just want to milk you for money.)
Please look after yourself op, grieve as much as you need too xx