Why does this happen in the first place?
Because you answered once. Every business that grew out of doing good work personally has this problem: the personal number was the business number, and nobody told the clients it stopped being one.
It is worth naming what it costs, because it is not only evenings. It is the mental load of never being off, the bookings lost in a thread you meant to reply to, and the fact that nobody else on your team can see any of it.
What actually works to move people across?
Give them a reason that benefits them, not you. "Message this number and you will get an answer straight away, even when I'm with a client" moves people. "Please respect my personal time" does not, however fair it is.
Then make the new number obviously better: it replies faster, it can actually check availability, and it does not depend on whether you are awake. Clients follow the channel that works.
What do I say to clients who ignore it?
Reply from the business number rather than the personal one. Do it consistently and the thread migrates itself, because people reply where the last message came from.
For the few who keep using the old number, a short friendly line each time — "sending this over to the salon number so the team can see it too" — does the work without a confrontation.
Does a business number mean losing the personal touch?
Not if the replies still sound like you. What clients value is being answered by someone who knows their name and their usual appointment, and that survives the move if the new number knows those things too.
What they do not value is a menu tree. If your business number makes them press options, they will go back to texting you personally, and they will be right to.