from full stop I hear "liquid" sloshing/gurgling from behind the dashboard

Clear the drains by the wipers. They are most likely clogged

Hard to say what that sound might be, could be vacuum leak, low coolant level causing sloshing in heater core. If you down load our free app. from Google or Apple, there is a recording feature that lets you send us an audio file of the noise. search for Bradley's Auto Service app.