I'll try to add to the conversation with 5e rather than suggest another game like everyone else (although I will say I've gotten into 4e recently and it's surprisingly fun with very tactical, engaging combats lol). I agree with a lot of what you said about sprinkling in descriptions and dialogue throughout the combat. I also think better designed enemies can help a lot. I highly recommend the Flee Mortals book by MCDM. I used it a lot in 5e campaigns now. More interesting enemies can make comabt more interesting.
I also think more interesting environments can also help. Put in lots of cover, spots of difficult terrain, maybe a trap, things like that.
Uninteresting combats should be fast, and one thing that helps that out is low health, high damage enemies, which for me means every now and then I lower enemy health if they already got a few good hits in or an enemy blew a spell slot to take care of them, since the point of this combat is to make players lose resources, not really for the story. This is less of a problem for low level combat, but more for higher level combat when health totals increase. Also, for these don't be afraid to make them some run to warn other allies, or run in fear if they get low health (unless they're mindless undead or constructs), or some run when their boss dies, things like that.
On the other hand, interesting or story important combats I'm okay with being long as long as they're also compelling and fun to play. After all, D&D has a lot of fun combat options in classes and players want to use them. That's where I recommend things like villain actions in that Flee Mortals book or the other stuff I recommend above.
Oh and the last thing I did is I am very liberal with consumable magic items or temporary buffs as awards for quests and stuff. I like the latter because players aren't encouraged to hoard it and it in fact encourages pushing ahead more to take advantage of the buff while they have it so the opposite mentality of hoarding, but the former has the nice advantage of being an additional option for a player to use in combat, which can make it more interesting, especially for those without spells. But to prevent hoarding, you have to make combat harder and have the kinds of players that look at everything on their sheet, as well as give enough rewards they know they'll get more in the future.
As someone who's only seen the first Alien and part of Alien 2, I really enjoyed it.