I realized that the tire rotates around the rim. Isn't it? So the balancing ofrim does not make sense. For the start change your bad wheel on spare. Do you have spare wheel in your trunk?
If the noise and shakes is gone, then the problem is clear.
milesvinsonhas right thing about rim! If rim have damage, and eliminate damage balancing does not work, then change the rim.
Remember, maybe you can hit the big holes and damaged wheel?