  In this paper we consider an approach to improve the performance of
exponential Runge--Kutta integrators and Lawson schemes} in cases where the
solution of a related, but usually much simpler, problem can be computed
efficiently. While for implicit methods such an approach is common (e.g. by
using preconditioners), for exponential integrators this has proven more
challenging. Here we propose to extract a constant coefficient differential
operator from the semilinear advection-diffusion-reaction equation for which,
in many situations, efficient methods are known to compute the required matrix
functions. Both a linear stability analysis and {\color{black} extensive}
numerical experiments show that the resulting schemes can be unconditionally
stable. In fact, we find that exponential integrators of Runge--Kutta type and
Lawson schemes can have better stability properties than similarly constructed
implicit-explicit schemes. We also derive two new Lawson type integrators that
further improve on these stability properties. The overall effectiveness of the
approach is highlighted by a number of performance comparisons on examples in
two and three space dimensions.
