shall I?; would you like me to?
Polite volitional form used to offer to do something for the listener. Often implies 'if it's okay with you' or 'shall I do it for you?'.
See also: ましょう
荷物を持ちましょうか。
Shall I carry your luggage?
shall we?; let's (do something)
Polite volitional form used to suggest doing something together. Softer and more inviting than the plain volitional form.
そろそろ行きましょうか。
Shall we get going?
ませんか invites the listener to do something themselves (一緒に行きませんか), whereas ましょうか offers to do it for them or proposes doing it together as the speaker's own action.
Polite volitional form of the auxiliary verb ます, combined with the question particle か. The form ましょう derives from the classical volitional auxiliary う, and か turns it into a polite question or suggestion.