Yes, states should allow you to drive with your valid foreign license from India until you get a State license from the State in which you are residing. States typically allow a person with a valid foreign driver's license to drive on their roads for up to one year. If you also have an International Driving Permit (IDP) it may also help since it translates your foreign license into English which is helpful to law enforcement and other state agencies that may have to try and read your license.
Whether your son's insurance will cover you while you drive his car with your international Indian driver's license is up to that specific insurance provider and their guidelines. So to find out if you will be covered by your son's policy he will need to call and check with his insurance agent.
To get a car insurance quote once you have obtained your US driver's license, follow this link.