I'm just guessing here ... but could part of the problem with your mom's attitude towards Shadow have to do with the fact that Shadow is your dad's dog, your mom and dad have separated, and your dad's dog is still living with you? Perhaps your mom feels some resentment towards your dad, and therefore towards his dog? Even if she doesn't say it, or doesn't even realize it, maybe it is subconsciously there anyway.

I kept my ex's dogs for a month or so, because he just couldn't get his ducks in a row. I did it for the dogs, and for his grandfather, who was stuck with them and was just too old to handle them. Yet, I did resent it, even though I volunteered to do it. Adult relationships are hard and confusing and ugly at times. We do honestly try our best not to involve our pets or kids in our problems, but sometimes it happens anyway.

I guess you are probably stuck with the pet care. You could try talking to your mom, making a schedule. That might work. But, if not, you'll just have to do it. It's the best thing for the dog. Maybe your mom just needs a little time to get her ducks in a row, too.