One of the most important lessons I ever learned in this life is -- "you can only control what you can control."
Not minimizing or invalidating anything, and I am well aware that there is no "off" switch for pain and anxiety, but scary as it may be at first, it can be strangely comforting and peaceful when you realize and accept that some things are just out of your hands.
Not to be too heavy, but, as a child I was forced to watch a parent die. I am the father of a dead child. I had a very close friend whom I was intimate with (just to illustrate the closeness) commit suicide after calling me on the phone and saying goodbye. I'm not telling you these things for pity; I'm showing you that just as your father has recently passed and now you face these circumstances with your dog, in my own experiences with similar darkness and tragedies I have experienced things that have profoundly compelled me to believe that there is much more to this life and "existence" than we may be aware of.
I am not religious. I am spiritual in my definition of it being that I believe there is "more," and in my experiences I am inclined to believe that death is not an end, but a transition. I don't believe that your relationship with your father is "over" nor has "ended," and I believe the same would be true for the connection with your pet. Energy can not be destroyed. I believe our connections with the ones we love, endure beyond and transcend even death itself.
And if you need something more practical to focus on, remember this truth: Your dog could have ended up with any number of people, in any number of places in any number of circumstances, but -- she ended up with you, and she is clearly loved. She clearly "won" in this life, no matter what happens now, and that is because of you; things could have gone a lot different, but you ensured she was taken care of, comforted, had all her needs met and lived it up as best she could have ever hoped for -- and she knows that and appreciates that. And hopefully there is still yet more of that life here to come.
None of us get out of here in one piece, man. Focus on what you CAN control, and don't get caught up in what is out of your hands. Don't worry about "tomorrow," just "be here now" and take the waves as they come, and know that you aren't alone in your pain.