Keep your dog at home. The only time my dog actually goes in a store with me is when I'm going to petco, because it's allowed, and the only time I leave him in the car is when it's something quick like going into the gas station to get a soda.
He use to cry when I left the house and while I was gone he'd be sad, and I'm guessing that's where some people have a hard time. I didn't baby him and take him everywhere because I felt bad. I left his ass at home and he got over it.