IF you have insurance on your current car, it automatically transfers to another car you acquire for 30 days. When you get a new car, the dealer will take care of all the paperwork. If they don't, then go to the DMV and they will help (but it will take a while). Likewise, if you rent a car, your current insurance covers you if you stay in the country. You can also buy insurance for the rental car.
You would technically be in violation of probation if pulled over, they ran you, and found you without an IID equipped. You face any number of violations and risk having the car towed if it's yours. There is no grace period for transferring install of the IID. You simply cannot drive without one if you are required as a condition of probation to have it.
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If found in violation of your IID requirement, you face a probation violation if the DA finds out about it. You also face criminal charges for driving on a suspended license. For DMV purposes, they can and probably will not allow you to drive with an IID, and make you ride out your suspension. They can also add on additional months/years to your IID requirement and suspension, depending on number of DUI and Driving on a Suspended license violations you have.
You need the IID on the new car before driving it. You need the IID on the car before test driving it. You cannot legally drive a rental car unless it has an IID. The only other car you can drive other than your own car (and your own car obviously must have the IID) is an employer vehicle. In order to do that you must carry proof in the car that your employer allows you to drive that vehicle without an IID. You can get the employer form from DMV.
Congratulations on your sobriety.