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You do not need weights on the inside and outside of your wheels. Weights should be placed on the inside or outside as needed (could be is either or both places). Your tire replacement mechanic should have taken care of this.
You can rotate tires front to back and see if that makes any difference. A front wheel imbalance will have a bigger effect on ride quality. If moving wheel/tire combinations around and you trust you tire change mechanic then you need to look for other hardware issues.
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