Well, it turns out that we really did have something wrong with our hydro-hot unit.
Talking with Vehicle Systems they suspected that somehow the fuel nozzle was damaged and the unit was running too rich. They recommended that we get the annual service on the unit which includes replacing the fuel nozzle and the filter and some general cleaning. This is also sometimes referred to as a "tune-up".
We had the service performed today. The RV shop said they could tell from the components inside the unit that it had been running too rich.
So far - no smell. Nothing like what we were experiencing before where we were overwhelmed with fumes - even with the slides in!
So - the moral of the story is that if your hydro-hot seems to be extremely smelly (diesel fumes, etc.), it's running too rich and it needs a tune-up. Also - if your exhaust starts to get a lot of black in it, you need a tune-up.
After we gain some experience with the unit under various conditions we'll decide whether to get the exhaust routed out the back.
Edited to add: We discussed our problem here: Hydro-hot too smelly!!!