I braze in my business (copper to copper and steel to copper). The coated rods have a shelf life. The flux coating will absorb moisture, crack and fall off over time, even if you keep it enclosed. I suggest only keeping as much as you will use in the next year or so. I use bare rod and dip.
Another problem is that there is too much flux. With copper or brass, there just isn't the problems associated with oxidation that you have with ferrous metals. Lightly coat the parts and the rod.
If you go to your local welding supply, you will find that they are a wealth of information and can advise you as to the proper rod and thickness along with the matching flux. I have my tanks filled at AirGas, nice company and knowledgeable about their products. They are nation wide.