Actually, this _never_ happens unless the nilradical is $0$ (i.e., unless the ring is a domain).  Let $A$ be a Noetherian ring with only one associated prime whose nilradical $N$ is locally free.  Then $A$ has only one minimal prime, so $N$ is prime.  Now localize at $N$.  We now have a $0$-dimensional Noetherian local ring $A_N$, so it is artinian.  In particular, $A_N$ has finite length over itself, and the length of the maximal ideal $NA_N$ is strictly smaller than the length of $A_N$.  But we are assuming $N$ is locally free, so $NA_N$ is free over $A_N$; the only way this can happen is if $NA_N=0$.  But, as you observe, every element of $A\setminus N$ is a nonzerodivisor, so the canonical map $A\to A_N$ is injective.  Thus $NA_N=0$ implies $N=0$.