I warn you in advance that this answer is a very long one.
Years ago, when my children were young, a one-year-old (or so) dog showed up in our front yard - out of nowhere, it seemed - and he just stayed. For years. The children took to him right away, of course. He had a collar and tag, and after several tries we got hold of the owner, who said, "Um... that sounds like my dog. I'll come over and get him" - and never came. By that time we were glad of it, because it seemed that his puppyhood had not been happy. Buster was big (I had never had a big dog before), and he was sweet, and he was troubled. When he wanted petting he would come to you - crawling on his belly. He had major problems learning where to toilet! He had trouble sleeping at night, but when I would let him out at midnight, he wouldn't come right back inside; he would curl up and sleep in the yard - even with a foot of snow on the ground. Does that tell you something?
The most difficult part, probably, was that he could not be left alone. Separation anxiety? Even in a different room for five minutes, there was separation anxiety. The children lost some stuffed animals. There was a nibbled-away place on the arm of the good sofa. My son's school project - a papier-mache eagle - was *eaten* when Buster was left alone and thought he'd been deserted. (My son made a new, improved eagle in record time, which turned out to be good for him.) We knew it was a knee-jerk reaction, not planned and definitely not malicious, but it was frustrating.
My husband said to get rid of Buster because he was so much trouble. And a woman agreed to take him. Buster looked back at us out the back window as the car drove away. He lasted there a weekend. The woman called and said, "He's digging under the fence and jumping over the fence, and going to the bathroom everywhere. Will you come get him?" When we picked him up he was so happy to see us it broke our hearts. Immediately we were in it for the long haul. And immediately Buster started going to the bathroom in the right place. (And eventually my husband got to like him.)
We taught him how to walk on a leash, though he dashed off without it if he could. We taught him some basic manners. At home, we checked on him and made sure he knew where we were. When I could afford it, I took him to basic obedience classes. We had to do two rounds of it before he even got the idea that he was supposed to listen to us. But after he caught on, he did better.
I never could get him to the point where I could leave him alone in the house. So he was put in charge of the back yard whenever we were gone. He had water, food, blankets, and appropriate toys. He bossed the squirrels around and played with anything that suited him, including the plants. Once, a few years later, when the temps were below zero, I thought maybe he had outgrown his anxiety and decided to leave him in the kitchen (what could he destroy there?), with expandable baby gates - the stretchy ones with the rather big openings - across the doorways. When I got home, I noticed that he had stuck his cute big muzzle through the gate openings and nibbled on the dining room rug.
So, in a definite way, we had to adapt to him; but he also adapted to us. He became more confident and easier to live with. His temperament was always so lovely that we were really glad to have him in the family. Those first few months were a struggle, but eventually he really felt more secure, although he didn't get over it entirely. When Buster was older, he developed a little nagging cough. It turned out to be lung cancer. He passed away fifteen years ago today.
I know your post is part vent. It always needs to be understood by every family that no matter who offers to help with the pet, or no matter who ostensibly owns the pet, the pet is Mama's. Even when my children were old enough to have fish or hamsters or parakeets of their own, it was Mama who supervised, who made sure the kids had done what they needed to do, to made the run to the vet's office (yes, there are vets for hamsters), and paid for those visits, too. It may not be what you bargained for, but there it is. It's okay to feel sorry for being put in this position, but don't allow any pity parties to go longer than five minutes.
Imagine this family member not as a dog but as a child who has come to you from a totally different environment. For all her life she has been treated with indifference, at best. (You know this.) She would like to be your best friend but, because of her fears, in a sense she doesn't speak your language. She doesn't know what to do. She's hoping you're her friend enough to help her. Or imagine her as old Great-aunt Millie, age 72, who has to move in with you because she has nowhere else to go. She's not cantankerous, but she's ill and depressed, and you're the only one she can count on. Will there be a light at the end of the tunnel? Yes, if you go to the end of the tunnel. Will there be a reward for you in this? I think so.
Talk to your vet. Pick his or her brains. Ask for an alternative to doggy day care (or a better doggy day care). Meanwhile, since you and the dog are a pair, talk to her, just as if you were talking to a friend. Sit down next to her and pet her for two whole minutes at a time (two minutes is a long time). Take her for short walks - just down the street and back. Do things that will get her more comfortable with you and perhaps a little more secure. When your children are at home (I don't know how old they are), enlist them to be care-givers for a few minutes at a time. Put washable blankets on the floors in all the rooms, so that the dog always has her own place wherever you are. You are right about not leaving an animal in a hot car; can you make arrangements with a dog-loving neighbor? Again, talk to your vet and see what options there are.
My eyes well up with tears to think that your family has adopted a senior citizen dog that nobody else has cared about! The adjustment that you're making now is different from the way you'd have to adjust if you'd adopted an eight-week-old puppy, but not really any harder. I think you can meet the challenge.