Do not cancel the old platform first. Run both for a few weeks: set up here properly, move your clients across, and only close the old account once nothing is still arriving through it. Cancelling early is how bookings go missing, because a client with your old link in their phone will keep using it.
Get your data out of the old platform before you cancel, not after. Client contact details, appointment history, notes and formulas. Most platforms offer an export and most make it awkward, and all of them stop offering it entirely once the account is closed.
Import your clients here, then tell them yourself. One message from you, with the new link, saying that this is where to book from now on. Send it to everybody rather than waiting for them to notice, and expect to send it twice.
Change the link everywhere it appears on the same day: your social profiles, your posts, your cards, anywhere it is printed. A stale link on a profile is the most common way a switch loses bookings.
Keep an eye on the old platform's calendar until its last day. Somebody always books there after you thought everybody had moved, and honouring that appointment is easier than explaining it.