[Tatsujin]

Hajimemashite means basically that you meet someone for the first time. Douzo yoroshiku is used in a greeting when meeting someone for the first time in a longer phrase. Hajimemashite, douzo yoroshiku onegai shimasu is the common "most polite" way to greet someone. Basically, and freely translated, it means just nice to meet you for the first time.

Yoroshiku onegai shimasu is the phrase you are looking for when asking for a favor. Basically it's just "please". Douzo is added for extra politeness, but you should use yoroshiku onegai shimasu, or onegai shimasu when asking for something. Douzo yoroshiku in a ask-for-a-favor-kindish -situation just sounds stupid, since douzu refers to action done by the addressed person.