It sounds like it's time for a small policy change. You could send an email to clients informing them of changes to your booking system, you should word it in a way that makes them realise that they are accountable for making their own appointments and that they are lucky to be coming to such a successful business. Terms such as "due to increased demand we are...", "to enable us to improve our service to you we have....", "to help you avoid disappointment you can now...." . Most importantly, you will need to implement a strict late cancellation policy as a result of this as this will prevent people from booking far in advance and then cancelling at the last minute.
If clients are getting angry about this change you booking system and let them know you've done it to make them happy. 85% of my clients book two to three months in advance when they are leaving the salon and they like that we advise them of upcoming staff holidays and busy periods so they can get what they want, when they want it. My clients know if they want the prime slots they need to book four to six weeks in advance and if they are looking for cancellation appointments for the weekend to call at 1.00 pm on a Thursday as that's when clients cancel for the weekend, but that's now a rarity!.
When I became that busy, a few clients became angry with me as they didn't like booking ahead, they liked to call on the day, I have lost them as we are no longer their type of salon. I simply attract clients who love what we do and book us in advance to get the service and quality they love.
Be strong, kind and firm, you're doing well.