Pannier shopping bags are made by
carradice,
Basil, and several other good brands.
The shopper style of pannier is a simple design with one large compartment and plenty of capacity for overloading at the top. A good touring pannier can serve a similar role.
If your panniers are bouncing off when fully loaded that is because they are not very good. A modern pannier mount has a rail , a couple of sliding hooks which support the weight, a rotating lock mechanism and an anti-sway hook at the base.
Any mount without locks, where the hooks (unlocked) cannot support the weight, or where the lower mount has elasticated hooks or velcro is of the old fashioned variety.