Is it really different from "no, thank you"?
No, it is just a little more informal than your version.
The audio does not sound correct.
The audio has a klingonian accent to be honest ! I'd like to improve something here !
"Nu, multumesc" ar fi mai romanesc!
”Mersi” is correct https://dexonline.ro/definitie/mersi
The Romanian version of ”mersi” would be ”mulțam”, but is used rather seldom.
There are lots of French words which are already part of the Romanian colloquial and even official vocabulary.