“…My husband (who's an expert skier) says I've got the technique right….”
Not completely correct: you obviously have not gained the technical ability, and thus the confidence, to “finish” your turns in a manner that will control (reduce) your speed to a level you can tolerate. You have little confidence in your ability to control (reduce) your speed, and that is typical of adults who are trying to learn to ski. Low levels of confidence cause high levels of anxiety and fear.
“…I've had 3 years of lessons and not progressed as much as I thought,…”
If this is the case, then there are two likely reasons: 1) apparently none of your instructors have recognized the importance of confidence-building “completion of turns” and therefore have not tried to teach it to you; or 2) they tried but have not been able to train you to do it.
“….is it the case of the more I do it the easier it becomes,….”
In short, yes, provided you slowly increase the activity of which you are fearful. To avoid being overwhelmed with fear as you practice going out of your comfort zone, you must make only small incremental increases in doing what scares you. Psychologists call it Exposure Therapy: deliberately exposing yourself to a small part of the feared activity, repeatedly, to get used to it. Rusty referred to it as “One of the techniques is sensitization….” Actually, you desensitize yourself.
“…makes sense about making turns more of an "S" than a "C"….”
The snake-like “S” turn shape is likely a large part of your problem because the middle of the “S” is where you are gaining too much speed. If you can be trained to make a “full”, rounded, “C”-shaped turn, you will be turning more across the hill and thus using gravity to slow down. The “S” has you pointed downhill too long, accelerating all the while. By contrast, the “C” has you pointed downhill significantly less time, so, less speed is accumulated. If you like, remember that “C” stands for Confidence.
“ For most people, speed control comes at the beginning of the turn -- "My speed is where I want it, so I'm going to rush through this so that I don't speed up". Sound familiar? For experts, speed control comes at the end of the turn; if needed, experts will turn uphill at the end of a turn to go slower...
The end of the turn is where you first need to build confidence – confidence that you can survive the acceleration that just occurred a second or two ago when the tips were pointed downhill. Unfortunately, your 3-years’-worth of instructors were unable, or did not attempt, to teach you how to confidently finish a turn, that is, how to end the turn to control (reduce) speed. Unfortunately, of the many instructors I have observed at least half blissfully spend great effort and time teaching a beginner how to properly start a turn, and they cannot understand why the client falls apart in the shaping phase of the turn and often falls. They apparently do not realize, or appreciate, that the shaping phase of a turn (the middle section of the “C”) is the freak-out zone for the client due to the rapid acceleration. Those instructors just don’t get it. The ones who do get it teach the finish/end of the turn first, and then teach the start of the turn later, after a satisfactory finish is learned, which provides the confidence to keep the client from falling apart in the freak-out zone.
I hope you can find an instructor who will teach you how to complete/finish the end of your turns in a manner that will control your speed enough to give you the confidence you seek. That instructor will probably have you doing a graduated series of “J” turns as JASP described in post #19. Likewise, there are many instructors who can introduce you to the concept of ‘skiing the slow line fast’; use an instructor who knows that. Also Known As controlling speed through turn shape. “Having a plan before things develop on their own accord…,” as Rich says, is what you want: have a plan for completing/finishing your turn (controlling speed) every time you start a turn.
Learn the completion of the turn and you will lessen the fear of starting the next turn.