Aviation Funding Stability Act of 2019 This bill provides continuing appropriations to the Federal Aviation Administration (FAA) from the Airport and Airway Trust Fund if an appropriations bill for the FAA has not been enacted before a fiscal year begins or a joint resolution making continuing appropriations for the FAA is not in effect. The bill provides appropriations at the rate of operations that was provided for the prior fiscal year to continue programs, projects, and activities for which funds were provided in the preceding fiscal year.