I live in a country cottage with a gable roof & cathedral ceiling, built originally for seasonal use. The ceiling is exposed rafters(full 2x6) with sheetrock in between attached directly to 1x10 roof boards. Asphalt shingle roof on top. To insulate, can I put ridge insulation up, then sheetrock over the ridge isulation?

You would need to fill the entire rafter depth with rigid insulation to get enough to meet code, so then you may as well sheetrock over everything. You could also screw more wood to the bottom of the rafters, and install fiberglass insulation with a ventilation space above it, but then you would need to create soffit vents and install a ridge vent.