My opinion:
Dampeners cause what is refered to as a "fizzle" they futz with the nuclear binding force to change it's profile.
So if a fission warhead is clipped on the way in, it "prematurely" cases extensive fission - the materials heat up and send out lots of radiation locally - but well short of chain reaction. Later is the "trigger" charges go off, the materials are no longer very pure (There has been lots of fission, so it is no longer pure plutonium/uranium) and chain reactions don;t happen.
So fission warheads could be salvaged for the plutonium/uranium - which could be re-purified and turned into a slightly smaller bomb.
If the Trigger charges didn;t go off, the casing itself will be heavily radioactive, however the Very high explosive could be salvaged.
For Fusion weapons, The Dampeners decrease the strong magnetic forces making fusion less likely, so the "magic" temperature/pressure that needs to be reached for fusion is higher, much higher (ie too high for "weapon grade" fusion weapons)
If a fusion weapon is triggered while in the focus of a Dampener, it destroys itself. Basically the explosions are violent enough to simulate the inside off the sun (however the dampeners mean that even this isn;t hot enough). SO you end up with a cloud of debris.
In either case, if the weapon fires, the explosive that "Triggers" the conditions that make either fission or fusion possible is highly likely to wreck the missile.
Alternatively, with decent control, a nuclear weapon may just be a nuclear damper itself - It Pushes the special materials the "wrong" way from Dampening so a lump of "normal" steel undergoes fission! or some cold Hydrogen starts Fusiing. That would explain the low cost and enormous power of the trav fusion plants. If this is the case, I have no idea of the effect of a dampener