-dAn is the shorter version of -dAn önce (before). It also means without.
Parayı ödemeden (önce) bir telefon edeyim. - Let me give a call before paying. (Note that it's informal to omit önce)
Parayı ödemeden lokantayı terk etti. - He left the restaurant without paying.
As you guess before and without are more or less related to each other when using them with verbs.