Since very few of the Civil - War era "Parlor" guitars seem to have used straps (from what I can determine from examination of period images and surviving instruments) I tend to play mine around camp sitting on whatever I can find or cross - legged on the ground. As an alternative, I have been known to find a stool, nail-keg, ammo box or such about knee - height and cock my left foot up onto it, resting the waist of the guitar over my sort-of horizontal upper leg (they seem to have always played over the left leg back in the olden days, and sometimes even used a brick or little "cricket" stool under the left foot while seated). This makes for a pretty stable playing position sans strap.I asked A'nt Mahtha, who just happens to be a Registered Occupational Therapist, for suggestions; she sez she really can't diagnose you without seeing, (and probably poking and prodding) you. It could be bursitis, or a weak scapulea (helped by certain excersizes), arthritis or any number of things. I have had my neck go out and cause all sorts of spasms and pain in my shoulder and arm, and about crippled me right up.
My left arm and hand has a bad habit of tingling, "going to sleep" and going numb after 4 or 5 songs. Usually I switch to the whistle or flute, do somthing a capella or tell a joke / yarn while doing some stretches A'nt Mah'tha showed me for "Thoracic Outlet Syndrome", which help quite a bit.
Nancy's advice to seek medical attention seems very appropriate. I might check with an Osteopath or someone who specializes in sports Medicine; refferral to a Physical or Occupational Therapist for specialized evaluation and treatment might be a good call as well.