"Shall we dance?" can be paraphrased rather easily; transforming it through use of a less archaic verb form than "shall" cannot be performed so easily. "Might we dance?" is a possibility, though "(S)hall we dance?" still seems current.
Person A: Shall we dance? Person B responses: Let's Dance. = good; We shall dance. = odd; Let us dance. = only slightly odd, though somewhat formal; Shall we, indeed. = intriguing, notably with respect to the level of enthusiasm intended as well awareness of the oddness of the word "shall"; Let us. = practically reasonable.
Might we indeed?
Teach Yourself language book series often made UK English words infrequently used in US English seem salient; use of "shall" with "I" or "we" instead of "will. Another example of Teach Yourself seeming British would be its tendency to indicate French cognates/grammatical contrasts in books for learning languages other than French. The cognates were often indicated using "cf." It even made me think that "cf." meant "cognate of" rather than the Latin for "compare to".