Is Eastern TN one of the densest populated regions for the GOP in the country or is that just how I perceive it from the shapes of the districts?
Eastern TN has quite a few midsized cities--the Tri Cities in the far East, and Knoxville and Chattanooga closer in to the center of the state. Both are extremely Republican relative to most cities their size, so yeah. That said there's still a lot of rural areas in Eastern Tennessee.
Why are these cities so Republican - and seem to be trending that way as well. I know they were historically Republican, but do they not have any demographic appeal to Democrats even with the shifts we are seeing post-Trump?
Most cities shifted towards Trump though in 2016, Clinton only ran ahead of Obama in metro areas with 1 million plus people, in plenty of smaller cities, including those with a few hundred thousand people there was a big shift towards Republicans, in the Trump era in 2016 at least only metros with 1 million plus moved towards democrats, not smaller cities.