Most bromine systems are two step. When you first set up, you're supposed to balance your TA and PH first, then add enough granules for a fresh start to begin your bromine bank. Then you add the floater and shock 1-2 times a week
If your tabs are slowly dissolving it means it's being eaten faster than it can stay in the water. You should check your PH levels. If they are not between 7.2 and 7.8 they struggle to effectively sanitize.
Honestly it just sounds like you need to use shock.