Am I using packing tape wrong?
Before cutting, you need to flip it in a way that sticks the tape to those little ears.  

It also helps to use quality tape. If the tape is crumpling and tearing when you try to peel it off e roll, that is definitely garbage tape, and just as it's side-tearing, it won't hold your boxes together either.  

The apocryphal source for quality tape of any kind is almost always 3M.  

It also helps to use quality dispensers.  Which those free red plastic dispensers found sonic-welded around the tape roll are definitely not.