To ‘shilly-shally’ is to be indecisive and vacillating.
It seems to make sense that ‘shilly-shally’ is just a variant of ‘dilly-dally’. The two expressions are almost identical and mean much the same thing. However, that isn’t the case. Although ‘shilly-shally’ is the later phrase in origin it was coined independently from dilly-dally’.
‘Shilly-shally’ is a colloquial way of saying ‘Shall I? Shall I not?’ and, as such, has more in common with the earlier expression willy-nilly (Will I? Will I not?).
Almost all of these ‘this way or that way’ expressions follow ‘this’ and ‘that’ in their construction. That is, the second word replaces the ‘i’ of the first word with an ‘a’. Examples are, dilly-dally, wishy-washy, chit-chat, zig-zag.