Is $a+b$ a unit if $a,b,a-b$ are zero divisors?

Question

Let $R$ be a commutative ring with $1$ such that each element is either a unit or a zero divisor. Let $a,b,a-b$ be zero divisors. Can $a+b$ be a unit or must it already be a zero divisor?

Answer 1

Try $a=2$, $b=5$ in $\mathbb Z/30\mathbb Z$.

From computational evidence, it seems that the $n$ for which we can find $a$ and $b$ with $a$, $b$, $a-b$ zero divisors and $a+b$ a unit in $\mathbb Z/n\mathbb Z$ are exactly the $n$ that are divisible by at least three different primes. See oeis.org/A000977. I'd love to see a proof.

Empirically, the smallest prime divisor of $n$ works for $a$, but the next prime divisor of $n$ does not always work for $b$.