I'm not car-free, but I do all my grocery shopping without a car. In fact, I'm at my local grocer as I type at 1.7 miles away taking 250 feet elevation gain and 210 feet loss. I understand your pain!
Here's why I still think a bike fits your situation. Namely an inexpensive folding bike with a front basket and rack.
Hills are conquered in the same way as a cart: walking uphill. Also known as "hike a bike." Folding bikes are usually also allowed on busses, so you could take a bus one way. You could time your departure to a bus schedule and shop knowing you don't have to rush or spend a long time at the store.
Folders can be brought inside and consume about the same space as a folding trailer.
My overall point is a folding bike is a trailer that you can ride downhill in. Electric would be a nice upgrade, but it's not necessary.
Specifically 5, 10, and 15mhz AM. There are others, but you'll really hear NIST WWV/WWVH if you're in North America/Pacific.