1) Do you feel D&D combat is slow (or "drags")?
Yes. Unquestionably. I don't know how anyone could reasonably say otherwise. Relative to other systems I struggle to think of anything slower.
2) If yes, how do you address this in 5E?
Play a different system. I'm sorry, but there it is. If you want 6 hour combats, and many people do, then D&D is great. It's all a matter of taste.
3) Has it always been that way? I'm not familiar with very much of 1E or 2E.
Earlier editions had faster combats. I personally prefer BECMI's speed. D&D's combat has slowed with each subsequent edition because it has gotten increasingly detailed, and those details add up in time spent.
//Panjumanju
That was not my experience with the previous couple of editions*. Possibly going further back, but honestly I don't remember that long ago. I could always devise a system that's faster. Play rock-paper-scissors to see who wins for example would be much faster. Or play 3.5 at mid-to-high levels with a caster that knows what they're doing and just have everyone delay until the caster goes.

Different strokes for different folks and all, but I find 5E combat a reasonable compromise most of the time and we have a lot of fun in and out of combat.
*In 4E hour long rounds were common at higher levels.