Adding dry ice to the water is a quick way to add more carbonic acid to the system. But the flask should be sealed to increase CO2 in the headspace, so more dissolves in the water over time.
Adding dry ice just increases the CO2 pressure to 14.7 and an equilibrium conc. will be reached. If the flask is sealed it will most likely explode. Any drift seen on the pH meter is drift in the glass and reference electrodes.