When the roads turn to ice or we have car troubles, we do walk to the grocery store (3/4 mile). Last year was the worst, and twice people screamed at me from their cars to get on the sidewalk. I was appalled at the rudeness from a town where usually when I walk people stop and offer me a ride! It is impossible to walk the sidewalks in icy conditions because half the streets don't have them on both sides of the streets, in spite of city codes many don't adequately shovel, and even when they do, the snow plows shove 3+ foot snowbanks to climb over at every intersection. Grrrr. Sorry for the venting, I just get frustrated with people who refuse to give walkers respect.
But my big thing is that if you're doing daily walking to the grocery store (rather than every few days), a really good backpack works best. Pack it yourself (a good idea even with the pull-along baskets, which we used in other communities where walking to the grocery store was an option). That way you make sure your heavy stuff winds up under your fresh veggies, bread, chips, etc. If there are two compartments to your backpack, you can separate glass items so they don't rub or clang.
I haven't used one of the carts shown in ages, but in retrospect, I'd buy a liner that you can with some or rig one, and not use the grocery bags at all. When hubby used to do it, he trained the bag guys to put two paper bags in the bottom, load the heavy items into them, close them, then two more paper bags and fill with all the lighter weight items, then any crush-able items on top.
One thing folk don't get is that I like this lifestyle. Everytime a business moves from town to the new 'burbs is one more time I consider moving, as much as I love this community.
Good backpack, which you'll use for other things in the future. If you think it is going to be a way of life, one of those carts. Pack the groceries yourself, or develop a relationship with the personnel at the store. Heavy on bottom, light on top. Glass separated.