I've been doing vision therapy and it didn't make anything worse. Temporarily you might feel strain like you do on a "bad" device; I take this as a sign the exercise is working. Unfortunately my progress has not been as good as others like @martin, and I will be looking to get prism glasses soon.
You should definitely see an optometrist that offers vision therapy. They will be much more helpful than ophthalmologists who typically just prescribe eye drops for dry eyes. A good examination will last at least an hour. You should ask if they are familiar with binocular vision dysfunction and esophoria/exophoria, convergence insufficiency, convergence excess, cover test. If they don't, find another.
You might connect with @martin -- I believe he is able to do some testing over Skype, but finding a proper optometrist is still critical.
There is a point in getting a diagnosis as the treatment: prism lenses and/or vision therapy may help you with your symptoms.
I had been to several eye doctors who were totally useless.