Il faut que requires the subjunctive mood. Il faut que nous partions. You would not add "me" here, simply the impersonal "il faut que" + subj. + subjunctive verb. There is some great information in the Tips and Notes for this section.

Partions is the subjunctive form of partir: don't worry if you haven't learned what subjunctive is yet, there is a lesson dedicated to it towards the end of the tree. Basically it's a special verbal mood required after some expressions including "il faut que".

Note that in this sentence, you don't have to use subjunctive with 'il faut': for instance, an equivalent (but a bit old-fashioned) wording would be "Il nous faut partir" where nous is an object pronoun just like in the construction "il me faut" which means "it is necessary for me [to do smth]".

And since 'nous' has already been mentioned in the previous sentence, you don't need to add it as it's already clear through context that 'nous' is the subject of 'partir', and so you can say "Nous ne pouvons pas attendre, il faut partir maintenant" which is the most natural-sounding solution to me.

"Il faut que" requires the subjunctive mood. The conjugation for that is "partions".

Partir = to leave
Sortir = to go out

"Nous n'attendons pas" is the translation for "we are not waiting" or "we don't wait".