The seasons of October and November are best, but everyone knows that and they come to Rajastan, so everything is costly too.
The off season, May to September is actually quite nice. Reason being the cheaper prices of accommodation and other activities. You ll have to avoid going out in the peak heat of the day, but the morning and evening shall be nice. You might experience some desert rain too. In the rainy season, cloudy skies make it pleasant even in the day. It is a risk though, with all the heat and rain. Proceed as per your physical conditions too.
The best time is between November to Mid March. You can visit now until mid March. I recently visited Pushkar and and spent two day there. You can refer to the detailed Pushkar guide here - https://bulbulonthewing.com/top-things-to-do-in-pushkar/
Best time to visit Rajasthan is from October- February, when the temperature is not too high. The days are sunny, not too humid, with cooler nights. https://www.tripoto.com/rajasthan/tour-packages