There are trivial examples that arise just by taking products of irreducible Einstein metrics with different Einstein constants. Whether an irreducible example exists is a much harder question. I do not know the answer to that, but I can think about it. (However, see below, where I answer this question.)
Of course, no such irreducible example can be Riemannian, and I guess, from your statement about the Lorentzian case, it can't be Lorentzian either, though I don't see that immediately.
In light of this, I guess the first case to try would be to see whether or not there could be one in dimension $4$ that is of type $(2,2)$. (See Addition 1.)
Addition 1: Indeed, there is an irreducible example in dimension $4$ of type $(2,2)$. Consider the $6$-dimensional Lie group $G$ with a basis for left-invariant forms satisfying the structure equations
$$
\begin{aligned}
d\omega^1 &= - \alpha\wedge\omega^1 - \beta \wedge \omega^2 \\
d\omega^2 &= \phantom{-} \beta\wedge\omega^1 - \alpha \wedge \omega^2\\
d\omega^3 &= \phantom{-} \alpha\wedge\omega^3 - \beta \wedge \omega^4 \\
d\omega^4 &= \phantom{-} \beta\wedge\omega^3 + \alpha \wedge \omega^4\\
d\alpha &= c\ \bigl(\omega^3\wedge\omega^1+\omega^4\wedge\omega^2\bigr)\\
d\beta &= c\ \bigl(\omega^4\wedge\omega^1-\omega^3\wedge\omega^2\bigr)\\
\end{aligned}
$$
where $c\not=0$ is a constant.
Let $H$ be the subgroup defined by $\omega^1=\omega^2=\omega^3=\omega^4=0$ and let $M^4 = G/H$. Then the above structure equations define a torsion-free affine connection $\nabla$ on $M^4$ that satisfies
$$
\mathrm{Ric}(\nabla) = -4c\ (\omega^1\circ\omega^3+\omega^2\circ\omega^4)
$$
while both $h = \omega^1\circ\omega^3+\omega^2\circ\omega^4$ and $g = \omega^1\circ\omega^4-\omega^2\circ\omega^3$ are $\nabla$-parallel. Thus, $g + \lambda h$ is an example of the kind you want for any nonzero constant $\lambda$.
Addition 2: Upon further reflection, I realized that this example points the way to a large number of other examples, all of split type, and most having irreducibly acting holonomy as soon as the (real) dimension gets bigger than $4$.
The reason is that the above example is essentially a holomorphic Riemannian surface of nonzero (but real) constant curvature regarded as a real Riemannian manifold by taking the real part of the holomorphic quadratic form, i.e., $Q = (\omega^1+i\ \omega^2)\circ(\omega^3-i\ \omega^4) = h - i\ g$. (The group $G$ in the above example turns out to just be $\mathrm{SL}(2,\mathbb{C})$ and $H\simeq \mathbb{C}^\times$ is just a Cartan subgroup.)
Now, the same phenomenon occurs in all dimensions: Let $(M^{2n},Q)$ be a holomorphic Einstein manifold with a nonzero real Einstein constant and write $Q = h - i\ g$ where $h$ and $g$ are real quadratic forms. Then $(M,h)$ will be an Einstein manifold of type $(n,n)$ (with a nonzero Einstein constant) and $g$ will be parallel with respect to the Levi-Civita connection of $h$. Thus, all of the split metrics $g+\lambda\ h$ for $\lambda$ real will have the same Levi-Civita connection as $h$ and none of them will be Einstein. If the (holomorphic) holonomy of $Q$ is $\mathrm{SO}(n,\mathbb{C})$, then the holonomy of $h$ will be $\mathrm{SO}(n,\mathbb{C})\subset\mathrm{SO}(n,n)$, which acts $\mathbb{R}$-irreducibly on $\mathbb{R}^{2n}=\mathbb{C}^n$ when $n\ge3$.
(Constructing examples of non-split type might be interesting$\ldots$)
Addition 3: By examining the Berger classification (suitably corrected by later work), one can see that, if $M$ is simply connected and if $h$ is a non-symmetric pseudo-Riemannian metric on $M$ with irreducibly acting holonomy whose space of $\nabla$-parallel symmetric $2$-tensors has dimension greater than $1$, then the dimension of $M$ must be even, say, $2n$, and the holonomy of the metric $h$ must lie in $\mathrm{SO}(n,\mathbb{C})$. Of the possible irreducible holonomies in this case, only the subgroups $\mathrm{SO}(n,\mathbb{C})$ and $\mathrm{Sp}(m,\mathbb{C})\cdot\mathrm{SL}(2,\mathbb{C})$ (when $n=2m$) can occur if the metric is to be Einstein with a nonzero Einstein constant. Both of these cases do occur, and, in each case, the space of $\nabla$-parallel symmetric $2$-tensors has dimension exactly $2$. Thus, the construction outlined in Addition 2 gives all of the examples of desired pairs $(h,g)$ for which the holonomy is irreducible and that are not locally symmetric.
To make sure that we get the full list of examples with irreducible holonomy, we'd have to examine Berger's list of irreducible pseudo-Riemannian symmetric spaces for other possible candidates. (I suspect that, even there, the examples will turn out to be holomorphic metrics in disguise, but I have not yet checked Berger's list to be sure.)
The case in which the metric is irreducible but the holonomy is not remains, and it may not be easy to resolve with known technology.
