I had a similar problem when I first started sparring. If I was just doing kicking drills or line work I would feel and look very loose and my speed reflected that. But , when I started sparring I would tense up and my speed would suffer. These days before I spar I just bounce around heaps on the balls of my feet and do a few footwork drills and generally just bounce around and feel light on my feet. Ive found it translates straight into my sparring and I am heaps quicker and I dont tense up. Basically if my feet are moving heaps when I spar I dont tense up.
do a million reps on a target, the try again to stay tense.
I think over thinking (it's a gem of a sentence, I know! ) is part of the problem. when I started sparring I had to REALLY think what I was going to do and set it up carefully. Luckily my opponent was in the same boat so it all turned out ok. add a couple of years and a lot of the moves have automated themselves. you don't have to think so much and it all flows better, thus more speed.
STATIC FLEXIBILITY! I use static flexibility exercises. I do front splits to one side then to the other side. Side splits and Center splits.
Advanced kicks that require you to be very loose, relaxed, fast and agile will improve. My Jump spin outer cresent kick improved a lot. Its faster and higher. It doesn't look like I'm tensing up.
Ah, I just wrote my essay for my promotion test about this!
For me, it is about recognizing which moments in a technique require speed and power and which ones don't. For, say, a front snap kick - well, that's a giveaway, right? It has the word snap in it! Lifting my leg to the chamber doesn't require much power/tension and can be done in a relaxed (I wouldn't use the word loose, myself) manner. Even the very beginning of extending the leg doesn't. It's that last moment - the snap - that requires the speed and power. I call the more relaxed portion of the technique "toned alertness" - kind of an awkward phrase but I think it's correct. I titled my essay, "Relaxation, power, and the conservation of energy".
I practice each technique thinking about the stance, balance, movement and what portion of the movement and my body can be relaxed and what portion more powerful.