Years of experience of taking in second hand dogs here - I quickly gave up trying to deal with the sort of private shelter you apparently want to get a dog from. Many of them treat you with a degree of suspicion, you have to fill out five pages of paperwork, they insist on inspecting your home, charge huge amounts for what are now often unsocialized, unhousebroken, elderly puppy mill breeding dogs, and then have you sign an agreement that keeps ownership of the dog in their name and and allows them to seize the dog without notice or reason.
Yeah. Such a deal. No thanks.
Do yourself a huge favor and locate the nearest large urban animal control shelter in your area. Their requirements are usually basic and reasonable, their fees are lower, once you pass muster with them they won't bug you or demand the right to march through your house and call up your place of employment to check how much you make a year. They want the dogs to find good homes but aren't obsessive/compulsive over doing an FBI background check on you as soon as you walk in the front door of the shelter.
I avoid the private rescues like the plague. Had nothing but bad experiences with them. And I'm not the only one. One fellow I knew - bred some of the top show dogs in his chosen breed in the country and the owner and editor in chief of one of the top all-breed show magazines - was turned down by a local rescue because he (GASP!) actually bred dogs. Like, he had a Toy breed that averaged two pups a litter and he'd breed maybe one litter every two years and the few pups he did part with went to carefully selected pet homes. But he didn't pass inspection and he told them their rules were so stupid the Queen of England would not qualify for one of their second hand dogs. Which was true.
So skip the private rescues. They are mostly run by people who are hoarders at heart. Go to the animal control shelters as the dogs there are under a time limit and if they are not adopted by a certain number of days they are toast.