If I apply with a new insurance company before the tickets are on my record, will I be able to avoid an increase?
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Yes, but the fees will go up when they find out. One way to defer paying a higher rate will be to prepay for as much insurance as they will let you (this is usually a year). You could always do the same thing with your present insurer. Prepay as much as they will let you. If they ask why, say you like the savings you get from paying for a whole year of insurance up-front.