In answer to your question, yes you should definitely see your doctor
to figure out what is going on with this cough. Any cough that goes on for this long or is this debilitating should be checked out by a doctor.
There is no reason that arthrogryposis would cause a cough, however depending upon the severity of the arthrogryposis respiratory infections can be serious. This is because, if there is scoliosis
or involvement of the rib cage, this can lead to problems with breathing and put you at risk of doing worse if you get a serious pneumonia. For this reason, people with arthrygryposis should always be more careful when dealing with these things.
There are several possible causes of a prolonged cough like this, including an evolving pneumonia, sinus infection, or even just nasal allergies. All of these possibilities can be evaluated by your primary care doctor
with ease. You should especially mention to your doctor if you have had fevers, increasing mucus production, congestion, or any other symptoms you think are relevant.
You should also consider pertussis, which is an infection that immunity to often wanes in the adult years.
You should talk to your primary care doctor as soon as possible.