I've commented on this subject every time I see it pop up and will continue to do so because I feel like it can be comforting to those of us with pets to know what happens when you leave your pet family behind. I'll not bore you guys anymore with telling why I feel like what I have to share is pertinent -- just click on my screen name and search back through my posts. I'm not trying to hide anything -- just that part isn't as important as what is in the next paragraph or three. Ok ...
Your pets WILL miss you. But you need to understand that our dogs and cats (and some bird species and exotic pets) do not grieve us in the same way WE grieve when we lose them. The best way I know to explain it is our pets live "in the moment". We humans live "in the what-if, -was, -could have been".
Depending on your dogs' personalities and bonds with you, they will likely search for you for a bit of time -- My dogs and cats, both, spent some time looking for my husband last year after he passed unexpectedly. The animals that were not as bonded with him stopped looking after a few days. Some of them would still look for him at certain times -- like breakfast time because he was ALWAYS sharing his bacon with them lol, or the time they expected him to be coming home from work. But generally speaking, those behaviors in the less bonded dogs and cats stopped, for the most part, within a month.
The one cat and one dog who were "his" buddies -- sat in his lap while he watched tv in the evenings, actively sought him out when they wanted attention (needed outside, for example) or wanted to play or get petted -- those 2 had a much longer longer adjustment period. In fact, the dog I had to have euthanized the month after my husband passed -- he was elderly and it was time. This would have happened even if my husband hadn't passed, but I do wonder if he went downhill so quickly because his favorite human was no longer here. And then the cat the month after that -- chronic kidney failure and he had no quality of life left for him here. To say it was a rough 90 days is a gross understatement.
Neither of those pets EVER stopped looking for him. They slept in his recliner every night. They sat at the front door every evening waiting for him to come home. That said I don't ever recall seeing a pet pine away for their human over a long period of time -- like humans will sometimes do. In fact, if my dad had an older client and he knew their pet was going to be passing soon, he would often "find" a kitten or puppy that "needed a good home" and try to enlist their help with getting the suddenly homeless baby a new home. (He was a veterinarian.)
I did have a recording of him leaving me a voicemail on my phone. If I played it, they would perk up and start looking around the room for him. But they appeared to be much more "triggered" scent than by sound.
There have been studies that have shown that both dogs and cats remember, and respond to, "their people" several years after the last time they saw them -- like if a pet got lost and then was found a few years later, or surrendered to a shelter or another family, and then sees the original owner again.
Someone else already mentioned it but if you just cannot stay any longer, please do leave a handful of items that have your scent on them for your dog. And make sure your family knows which item(s) you have left for them and why.
Finally, I am so sorry you feel so awful that you even have to contemplate leaving your dog behind. But just know that as long as they have a loving family around them they will be okay.
And I have had pets for as long as I can remember. 
