I'm going to be switching to return customers only for the summer. Mostly, it's for the worst of allergy season, since both my partner and I have allergy troubles and it's always a risk taking in a new dog - occasionally one will just make us both miserable - but we know what to expect with our regulars. But also, we've been busier than we'd like for a good part of this year, and we're getting to the point where just our regulars are taking up plenty of our availability.
Between now and the 26th, we'll be hosting 6 dogs from 5 families with no breaks in between. All of them are return customers. We generally only take 2 at a time, though we'll make an exceptions for short overlapping stays with dogs we know well. So most days in the next couple weeks we'll have 2, but a couple days only 1 and a couple days 3.
If you're finding yourself frequently turning down regulars for new clients, I agree with the other suggestions here - raise your rates (and honor the old rates for prev clients if you wish, though when I've raised my rates, my existing clients have been happy to pay the new cost), send messages to your returning clients reminding them to book busy times in advance, and/or lower your available slots to keep one free for last minute return guests. All of those can help reduce the number of requests you get but won't be as drastic as changing your status to return customers only.