I think there is a difference between awkward and uncomfortable.
You can have an awkward appliance made for you that is comfortable and will create the effect you are looking for. Thicker acrylic on the palate will give you a bigger lisp and a tongue crib will do the same. Both can be comfortable.
Drooling is your body's response to having something in your mouth that it is not used to, and there isn't much you can do to change that.