Is it possible that the "negative energy" needed to create stable wormholes as theorized by Morris and Thorne (?) could be dark energy? Could this be the "exotic" negative energy matter referred to?
Your title asks about dark matter, while the question asks about dark energy. The two are very different!
I'd guess you mean to ask about dark energy because this does behave differently from matter (as far as gravity is concerned dark matter behaves just like ordinary matter). If so, the answer is that dark energy does not behave like exotic matter and cannot stabilise a wormhole.
The effect that matter/energy have on the spacetime curvature is described by the stress-energy tensor. This tensor contains terms related to energy density and terms related to pressure. Dark energy behaves as if it has a negative pressure, which is why it is causing the expansion of the universe to accelerate, but the energy density term from dark energy is positive just like regular matter. The exotic matter that we need to stabilise wormholes has to have a negative energy density.
Dark matter has the same gravitational effects as regular matter it just doesn't interact electromagnetically. We expected a certain amount of dark matter, such a neutrinos, bit there weren't enough neutrinos so it seemed like there must be something else besides things like neutrinos. Something that interacts in the normal way gravitationally but just doesn't interact electromagnetically.
Light is an electromagnetic wave so when something (like a neutrino) doesn't interact electromagnetically you can't see it with light.
We needed something like a neutrino but not any if the things we've seen before, there's just weren't enough of the things we've seen before to explain how massive the galaxies seem to be.
Exotic matter with negative energy densities would just make it worse then we'd need even more dark matter to compensate for the exotic matter.
Now sometimes people might say exotic dark matter to mean the new stuff, the stuff we haven't seen before to contrast it with neutrinos and such which would work fine but there just isn't enough of it. But they don't mean negative energy density that would do the opposite of what we need them to do.