Open mouth could mean mouth breathing usually because of nasal obstruction and possible enlarged adenoids. If this problem is corrected then your drooling problem should be solved.
Next Steps
Visit an ENT specialist to rule out any nasal obstruction may it be even a polyp.
Health Tips
Then get yourself examined by a dentist to check alignment of teeth and incompetent lips. This would be the right age to rectify the problem rather than ignoring it.