I don't understand what you mean by "grace period?"
You can file a lawsuit in small claims court for what you paid for the tires plus any additional monetary damages you suffered.
Auto manufacturers recommend not putting used tires on a vehicle at all and never using tires over six years old. A tire will have an identification number on the sidewall that begins with the letters DOT. The last four digits represent the week and year the tire was manufactured. For example, "DOT ANVF F771 1909," indicates that the tire was manufactured in the 19th week of 2009.
You might have a police officer look at them to use for evidence in civil court. If they sold you something that is illegal to operate on the road, then the tires are not fit for the intended purpose. Under Florida 316.610, a police officer may stop a vehicle that appears to be unsafe for any reason and do an inspection. If he/she deems the vehicle to be an immediate hazard, the vehicle can be removed from the road until the repair is made. If not, a citation will be issued and typically the court will require the repair as part of the adjudication.