No actually the pressure goes down, the coolant boils at a lower temperature and that's when it dribbles. 1# of cooling system pressure will raise the boiling point 3°. You loose pressure (15#) and you are back to water boiling at 212° (depending on coolant strength and altitude
The pressure in the cooling system stays at 13 psi (engine on or off) till the coolant is cool enough to start to contract and either suck air or coolant from a recovery bottle back in behind it.
I agree as it is cooling it gets below the pressurized boiling point of the coolant at 13 psi, but then it cooles to be below the boiling point of the coolant at 12psi etc till it is room temp and atmospheric pressure.
I have never seen a cooling system overheat and boilover after shutdown unless there was some other problem, like a leak, trapped air etc.
