Poochi7, you’ve already gotten some very good advice. The trick will be to get Snowy down and into a smaller, confined area to give you both the time to develop a better relationship of trust.
If she was stray or spent a lot of time on the streets, I’m not at all surprised that she’s hiding, full-time, in your house full of barking dogs. Stop and think what barking dogs are, to an outdoor stray… they’re excited, dangerous, and something to GET AWAY FROM! So Snowy’s trying to get away, but she can’t. So she’s just hiding as hard as she can, somewhere the dogs can’t get to her. From her point of view, she’s doing exactly what’s helped her survive outdoors.
Have you taken any steps to train your dogs to bark less? I know when one dog barks, it can set the rest of them off… but having them bark less will probably help your new cat adjust better.
In addition, if you can help ensure that the basement is somewhere the dogs don’t go, that will give her an area where she will feel safer. Do you spend much time in the basement? Do your cats spend much time in the basement?
In the rest of the house, do you have cat furniture or areas where the cats can get up high, away from the dogs? Her very first instinct, with the dogs, for a very long time will be to get away from them – even after you spend time with her in an enclosed room, to develop a relationship of trust.
When she’s confined to that room, after a few days or weeks, you might be able to bring in your most cat-safe dog (the one meekest, oldest, and/or most patient with cats, and least likely to snap back if the cat takes a swipe at its snout), to meet Snowy. However, keep in mind that your dog will need to be on a very short lead, and Snowy will probably not react well. Patience and repetition will be needed, in large supply.
(As much routine and predictable behavior as possible is needed right now, for Snowy to understand her new family.)
Is Snowy deaf? What color are her eyes? If she is deaf, that could add an additional layer of work to develop trust. (Even if she’s deaf, she can certainly pick up the vibrations in the air, when the dogs bark.)
Last, if it turns out that she’s not adjusting well to the rest of your family within 4-6 months, what will you do?