I do prefer a rack, but fyi if you're carrying all that stuff it's definitely going to affect the handling and turn of the bike on a pannier as well.

It doesn't really matter which rack you get if your bike has rack mounts. I just get a topeak rack, they're all pretty much the same unless you want a fancier top mounting system, which it sounds like you do not. This kinda of stuff is usually noticeably cheaper on amazon than from a bike shop.

There's a pannier/backpack called the Arkel Bug that's a pannier on your bike but a backpack when walking around. It's expensive though:
https://www.youtube.com/watch?v=zYEmp4toAck

For grocery shopping I have the Banjo Brothers Market Pannier. It's the only pannier I could fit a digornio pizza box into - it's the same size and shape as a grocery bag. For groceries, they've been fantastic. Their drawback is mainly that they're poor for aerodynamics, you can feel the drag while riding with them - a direct result of their grocery bag shape.
Amazon.com : Banjo Brothers Market Pannier : Messenger Bags : Sports & Outdoors