Hello and welcome to the forum!
The best time to visit Rajasthan is actually during the winter season. The months from December to March are the winter season months. January is the coldest month in the region. Temperature ranges from 20°C to 35°C. But it can also get as low as 0°C or the negative in other cities in Rajasthan. It's a great time for sightseeing as the weather is cool and pleasant. Unlike the summers which can get humid and hot or the monsoon season when humidity also rises. You only need to pack your jackets to prepare for the cold.
When it comes to the fog, it usually occurs during early morning. It is usually in Delhi and Agra that fog is quite bad. In some parts of Rajasthan, you might find them more foggy. Like in Jaipur, fog occurs in the morning and can stay for a few hours. During this time, you can visit other attractions first that aren't as affected by fog. By mid-morning, the fog would have settled and you can continue your journey.
It also depends on the weather itself for that day. Some days you'll encounter fogs and other days might not have fogs. And usually, during the third week of January, the fogginess might be gone. It's at its worst during the end of December and first few weeks of January.
So there would be no need to cancel your trip. The months you picked are actually the peak season for tourists. It's just a testament that these are the best months to visit Rajasthan.