Are you receiving more than one booking in a single time slot for a single member? It is important to ensure that the booking and availability calendar match up. To check this, log into Appointlet and head to Members > member name > Calendars and ensure that the Booking Calendar is also ticked in the Availability calendar section:
If the correct availability calendars are selected and you still have this problem, it's likely due to existing calendar events marked "Available".
You will need to make sure your manually created calendar events are marked "busy" to block those times in Appointlet.