I'm not sure what you mean exactly, but there is a badge you can use that says 'new' and you can set the time during which after a post has been added is will show. Combined with fields to which conditions are added it should be possible to create what you want.
It works fairly simple:
Add a badge where you need it, field key is 'post date', field condition is 'less than', value to match is 10. The field where it says 'badge' is where you can enter the text you want to show.
This way you have a badge that shows for ten days with a text you want. If you want a badge to show after ten days you should use 'grater than' as the field condition.
That part is a no-brainer as GD has it all built in. A matter of playing around and trial and error. I use a 'new' badge on one of my websites and it works like a charm.
But what puzzles me in what you ask is that a listing owner posts a listing which can be booked? If so are you using the Booking Marketplace for that? Maybe you can elaborate on that and maybe share a link?
