The above methods will work, but are slightly flawed. You can sous vide a whole quail, but it is inherently wrong to do so. The white (breast) meat is inherently more tender and requires less heat than the tougher legs and wings. Separating the breasts and wings/legs into two sous vide bags works the best.
I like to cook the breasts at 130°F (55°C) (hold for ~2.5 hours for food safety reasons) and the legs/wings at 140°F (60°C) for 12 hours. The latter process helps break down the tougher tissue in the leg and wing meat, providing a seriously tender piece of quail. Filling each bag with a good flavorful broth or stock such as duck broth will make the quails even better. Make sure to remove the skin prior to cooking the meat. Finish the meat off on a smoking hot griddle for a moment to crisp it up.
Sans sous vide, I would prefer to simply glaze and char-grill the quail. Braising is a great technique, but in my experience is better suited for less-flavorful cuts of meat. Braising is probably going to extract much of the flavor out of the quail (and make a delicious jus...). And it sounds like you're braising it too long if it's dry.