'I sure will' sounds good, but I think 'sure' needs to proceed 'will' because 'I will sure do' is strange.
So you could also say, 'I sure will do', but the exact meaning of 'do' has to made clear from the context.
For example
A: You take care of yourself!
B: I sure will do! ( do - take care of myself)
Otherwise, 'do' needs an object:
I sure will do it