What if the screw is tightened up inside the bathtub handle fully and the handle still wobbles?

If you are talking about the faucet handle then you may have a broken insert, stripped handle. Some tub and shower faucets have plastic inserts to grip the faucet nipple. This nipple can be square or ribbed. Once the corners or ribbed areas become worn they will not tighten up even if the screw does tighten. Look carefully at the nipple with faucet handle off and look inside the handle to see if the corners or ribbed areas are worn and check for a plastic insert , this could be cracked which will cause the handle to be loose.