Honey Bunny, you KNOW you aren't a bad mommy.
You are just mourning right now, and that is fine. You need to let it take whatever form it needs to, and you need to let it take however long it needs to. God, Cabo has been gone from me for almost 2 years now, and there are still weeks that I cry every day for missing him. Right after I lost him, I didn't even want to LOOK at Gwen, let alone pet her or pay attention to her...
It is normal right now to be focused on what you lost, and for your sadness to override everything else. This JUST happened. You loved him for years, you can't get over it in a day, and no one should expect you to do so. You need time to heal a bit. Pretty soon Grey will do something silly and make you laugh, and Zeus will come and plop his big ole' head on your lap and even though you will still be sad, you will be taken away from your sadness for a minutes, and things will feel just a bit better.
If nothing else, even just the fact that you are concerned about being a bad mommy shows that you are not...