When I first started out shopping by bike, I pretty much stuffed my backpack full. I had an MTB so a pannier wasn't really an option. If things ended up not fitting, my handlebars came to the rescue to hold the rest of the bags.
With my current bike however, things are a little different. The rear pannier rack enables a few more options and things to try...
Carabiner
Tried and tested, never failed me yet. Using the carabiner I can carry a pretty heavy bag on the rear pannier, without shelling out $$ for an expensive cycling bag.
The bag does stick out a bit depending on how packed it is, which is something to be aware of.
Motorcycle helmet clips
These mainly allow me to secure boxes and water bottles to the top of the pannier rack, while still being able to quickly attach and detach shopping bags from the side.
While the helmet clips themselves have been flawless, the bags themselves have been less than stellar - every single one of them have ripped so far ☹️
DIY pannier bag
I did consider getting a proper cycling rack bag... but they're a bit costly and can't really be used for much else.
After some research I made my own, using a reusable 'cooler' shopping bag from a supermarket, and a cheap pannier attachment kit. I had some short bolts, nyloc nuts and washers available to put it together.
It's so much more convenient to use than the other two methods above, but I haven't put it to the test yet with heavy shopping. The zip is a pretty nice touch to keep shopping cool and dry.
Additional pictures
How do you guys carry your shopping, or any goods in general?