Western has a somewhat different approach to following the motion of the horse's back than dressage does. The non-dressage person in me, who watches videos sometimes and reads books about dressage, thinks dressage rewards making the difficult look easy. The dressage position expect you to move with the horse by flexing in your lower back while keeping your feet positioned for lots of subtle cueing.
Traditional western riding is more like this:
Your feet are more forward - perhaps not as much as this guy's was, but some western saddles will force a chair seat - and you use the hinge of your waist to go with the motion of the horse's back. My personal experience is that a bit of a chair seat makes it easier for someone with a stiff back to move with the horse.
My only fall from a horse was shortly after I started, in Jan 2009. The injury to my lower back was all soft tissue, but I spent over 4 years riding like someone with a 2x4 stuck in his jeans. My lower back is finally getting a little looser, but it still doesn't have the flexibility and strength to make the dressage position work. However, if I adopt a position like the guy above, I can move in synch with the horse. It is not subtle, but it works for me & my horse.
This is a long way of saying that your western saddle may make it easier to move with the horse either because of the position it puts you in, or possibly because psychologically you are willing to relax more in a western saddle.
BTW - my experience trying to ride in a position like the above cowboy's is that it is surprisingly stable and easy in terms of distributing weight and following the horse, but that it sucks for giving cues with your heel or even your calf. Also, I didn't realize I was tending to lock my lower back. It was a subconscious decision, made to prevent pain and swelling. Like many subconscious decisions, it didn't work out the way my subconscious expected it to...
Good luck to you! Keep riding & keep positive!