It might be worth looking up de Quervain's tendinitis, which is what my sore, swollen thumb joint turned out to be.
My experience was that splint suitability is quite individual, and you might have to try more than one. I ended up with a hard plastic one that's mainly designed to keep the wrist straight – it somehow seemed to support my thumb better than anything else. A pharmacist may be able to advise if you can't get to your GP. Other treatments that helped were icing the area, and reducing as much as possible any activity involving that joint.
I do feel for you, though, it's infuriating! I hope you manage to find something that works.