OTOH, given that it is (like most of the stuff around it) orbiting the center of mass of the Milky Way Galaxy, shouldn't Earth be moving relatively close (within a very small fraction of light speed) to the speed of most of the interstellar dust, etc., in its immediate galactic vincinity, such that moving at 0.998c relative to earth is also moving at a very similar fraction of c relative to lots of other things you are going to potentially hit on a journey from earth to a nearby star system?

I believe that would be wrong. Also not an astrophysicist but I would guess that most of the stuff in our galaxy is moving at speeds which are, on average, relative to our galaxy, and moving at some average rotational speed around it. Of course there could be stray bits moving quite quickly on through.

So moving at c relative to the planets and solar systems swirling around our galaxy probably would make you more likely to hit something.