I am a western Washington rider, too, and I understand what you are experiencing well. Many years ago I moved my old Arab from California open country to the heavily-wooded Pacific Northwest. I struggled with him at first, but then I thought about it more deeply and changed my approach when I noticed that many horses I have known are generally less-comfortable in the woods than they are when they are in open spaces, some just handle it better.
Horses are creatures of the open plains, and are perfectly evolved to survive in flat or rolling grasslands where they can see for miles and run fast in any direction when needed. Many of the same systems that serve them well in that environment fail them in what feels like the confines of a forest. If you have a horse that is naturally more vigilant (and some just are), and/or less trusting, you are going to struggle with this more.
Horses are naturally far-sighted, so that they can see a potential threat a long ways before it is actually a danger, and decide what, if anything to do about it. They see fairly poorly, and do not have great depth perception, at close range. They also have good spatial memories about what should be where, but their brains (bless their hearts) are set up to map a few clumps of trees here and there in a mostly open space, not the infinite complexity of a dense forest that they can’t see all that well.
As a horse in the forest, there are a gazillion things that might be a threat, but it’s hard to tell (especially when you are being asked to move fast), and if you decide (rightly or wrongly) that something actually is a threat, it’s already close enough to “get” you by the time you figure it out. And then if you think you want to flee that threat, you may or may not have an open path to safety away from it. When I think of how horses experience that environment (especially the hotties), I can be much more patient with the ones that struggle to function in it.
Also remember that if your horse starts to feel scared in his new environment, and you inadvertently do something to reinforce that fear (like force him forward, or get scared yourself), you have just reinforced his basic understanding that this environment is really threatening, and he needs to be even more careful (what we call spooky) in the future.
Like others have suggested, patience is what it takes, and the ability for you to remain calm when your horse’s natural fear starts to take hold. It may mean you have a very slow, quiet year while you gradually acclimate him to his new environment using gentle, trust-building, ground techniques, then transitioning to baby steps on his back- especially since you will be working on your own fear at the same time. In my experience, these types of horses may always be a little edgier in the woods than they are in the open, but they can be ridden safely when they learn to trust you aren’t ever going to take them somewhere that is actually going to harm them.
I am not sure where in WA you are, but if you’re on the Kitsap Peninsula, perhaps we can connect in real life since it sounds like you haven’t found sympathetic help out here. Sorry! There are nice, patient riders in the area, you just can’t see them for all the trees…