Have you ever scanned a newspaper or magazine cutting at too high a resolution, the sort of res that's usually fine for scanning photos, but that makes a really grainy and fuzzy mess of newspaper print?
That's what a lot of TNG HD looks like. To a layman, it looks like either much of the film stock is in poor condition, or it's been scanned at too high a resolution.
Quite a few episodes I've watched from season 2 (OK, it's season 2!) have so much grain and distortion on the red of the command uniforms (usually Picard's) that the image seems to be on the verge of breaking up. That's the best I can describe it. Sorry.
I seem to be in a position now where due to the excess grain, I'm getting so disgruntled at the picture quality of a lot of these episodes, that I'm almost preferring the upscaled DVDs at 720p.
Well, considering the DVDs are objectively inferior in every possible sense compared to the Blu-rays, that's a pretty drastic thing to resort to... and so there's something either fundamentally wrong with your setup or your expectations. The film stock was actually in very good condition and it was not scanned at too high a resolution.
I don't mean this as condescending, but I assume you're hooked up via HDMI and not analog component cables, right? Have you calibrated your display -- at all? You're not viewing the episodes in a "dynamic" or "vivid" mode are you? If so, that could be unnaturally brightening the image and boosting the colors. Also, your sharpness should be set very, very low. You should not be artificially sharpening the image, that can make everything (particularly the grain) look very, very bad. And definitely turn off any DNR.
If your display is uncalibrated, start with a "movie" or "cinema" mode and tweak from there, based on the lighting conditions you're viewing the display in. That kind of a mode should be closest to a 6500 Kelvin or D65 reference color and have the best black levels. If you have a Blu-ray with a THX Optimizer, that would be better than nothing to calibrate with. They also have a free iPhone app that you can use called THX tune-up.