I will give a more extended hint, which should cover cases A) and B) simultaneously. To show that a sufficient statistic is not complete, we need to find a function of it with expectation identically zero. To this end let the usual sufficient and complete statistic be
$$
(\bar{X}_n, S_n^2) = \left(\frac1n \sum_1^n X_{(i)},\frac1{n-1}\sum_1^n (X_{(i)}-\bar{X}_n)^2 \right)
$$
which also shows that this sufficient statistic can we written as a function of the order statistics.
Now consider the centered order statistics, given by
$$
( U_1=X_{(1)}-\bar{X}_n, \dotsc, U_n=X_{(n)}-\bar{X}_n )
$$
Can you show that the expectation of $\sum_1^n U_i$ is zero? And what is then the conclusion about the completeness or not of the normal order statistics?