You can prevent double bookings in an Appointment field by connecting at least one Google Calendar to it.

When the form is loaded, Paperform will check to see when the connected calendar is busy, and it will automatically grey out any unavailable times. When a new appointment is booked on your form, this will create a new event in the connected calendar as well, so double bookings will be prevented automatically.

To connect Google Calendar to your appointment field, first click on the Appointment question in the form editor, then click Setup the Appointment in the right-hand side panel. The Settings tab will appear, and you can click + Calendar on the right-hand side (screenshot below.)

If you have added the calendar to a form before, it will appear in the dropdown menu here. To connect a new calendar, click the "+ Calendar" button to the right and follow the prompts to give Paperform access to it.

You can also connect multiple calendars to your Appointments field, allowing you to accept appointments until all of them are booked.