What are we doing wrong? Just trying to hang a picture [?]

My husband and I have some serious issues with the walls in our house. It was built in 1996; we live in CT. This is the 2nd time we’ve tried to hang something on a wall, and it has turned disastrous. We bought the self drilling drywall anchors recommended to us at Home Depot (TOGGLER SnapSkru). We used a philips screwdriver and once it gets to the threaded section of the anchor, it just chews up the Sheetrock. We are now left with a hole that we need to patch and try again but I’m afraid. We made sure there was no stud behind the Sheetrock. This is the second spot in this house that this has happened. Ace Hardware and Home Depot were no help. They recommended the same anchors again. I don’t know what to do and we’re having a screaming match over this. This should be easy, shouldn’t it?!