I am such a sitter in Portland, OR and I also work with a local dog trainer who deals with dogs with SA. I work as a traveling sitter who stays in the owners home and I get quite a few requests for dogs with SA from mild to severe cases. Since pet sitting is my primary source of income I am able to accommodate the needs to a pet owner who has a dog with SA, but because of the difficulty in long term stays when needing to do grocery shopping, etc if even for a short amount of time the price range for stays are custom tailored to the needs of the dog and the owner.
Both of these questions depend on the sitter. It sounds like you're interested primarily in travelling sitters who will come to your home and stay with the dog. You'll want to contact the travelling sitters in your area to discuss your dog's unique needs. Some may not be able to stay 24 hours per day, but some will be able to accommodate you. I'm not sure if you're open to boarding, but some boarders will also be able to stay with your dog for 24 hour care. From the travelling sitters I've interacted with, most don't take multiple clients at a time since the job requires staying at the client's home. Some do, but again, this is something you would want to address with the sitters in your area.
To search for available sitters, enter your zip code into the search field, and select from the options at the top of the results page to display only sitters offering the services you require. Then message anyone who appeals to you with the specifics!
Do sitters take a large number of dogs at a time?