There is little doubt that you will need an upper GI endoscopy with a 6 week history of ?spitting? blood. However, your description is less than adequate. Do you have any hoarseness or change in voice associated with your symptoms? Are you vomiting up blood with stomach contents even if liquid, or are you spitting up teaspoon-tablespoon quantities with surrounding saliva? Do you notice that blood or fluid seems to well up in the back of your throat and then you spit it out, or does a large amount suddenly seem to regurgitate into your mouth associated with prior nausea? The nausea could be because this is a primary stomach problem. Or because simply having blood in the mouth makes many patients nauseated.
The distinction is important because your problem could be as simple as bleeding from a dilated vein at the base of the tongue (Hemorrhoids of the oral cavity) or dilated vessels elsewhere in the hypopharynx. Being on Coumadin would allow this area to bleed more easily, where in a non-anticoagulated patient they might never notice any blood since it is easily swallowed rather than welling up in the throat.
Again the site of bleeding is important and the GI endoscopist will very likely miss seeing anything in your mouth or hypopharynx because the large tube used, does not focus closely as it goes down. At a minimum, ask the endoscopist to take as close a look as possible at your tongue and larynx on the way down (a suggestion from an ENT doctor).