Stop the car safely and open the windows or doors to let fresh air in as soon as possible. It is likely that the dry ice (solid carbon dioxide) has started to sublime and the resulting change to the atmosphere is causing the drowsiness. It is important to consider ventilation when transporting any cryogen with the ideal mode of transport being a vehicle with a separate cab. Even when restrictions on transporting chemicals do allow for them to be moved in a normal vehicle good practice should be followed such as travelling with open rear windows.
Charles Harrison is a safety adviser at a UK university