On most winter days, fog is not serious. On some days (unfortunately, mostly in early to mid January) it is very serious. You have been given worst-case advice since you really need to catch the intl flight.

It is quite possible that it will be a bright and sunny day and you will be in Delhi 10 hours before your intl flight for no reason. Rent a car with driver and have them take you to some of the sights. In the evening, have them take you to Dilli Haat, a collection of stalls offering local products from all over India that is open to 10pm.