I volunteer at the information booths at the airport every week. Let me add a few comments to the suggestions above.
There is no storage for luggage anywhere in the airport. This has been true at least since 9/11. While I agree that Travelers Assistance is very helpful (they’re the ones who run the volunteer program - some 500 volunteers and counting!), they are highly unlikely to agree to store luggage. The airport is understandably very sensitive about security and volunteers are instructed not to touch, move or open passenger bags.
You might call the Intercontinental Hotel to see if they would store a bag. You can walk to this hotel if you do NOT leave the secured area of the airport. Follow the signs to Gate C25 (some of these signs will also show the hotel’s name) and go across the skyway (bridge) to the hotel. If you happen to leave the secured area, you would need to follow signs for hotel shuttles on the ground transportation level and take their shuttle. To return to the check-in area from the hotel, it’s best to take their shuttle.
I go through security a lot and the lines are rarely longer than 15-30 minutes, usually much shorter. International flights begin boarding about one hour before departure, domestic flights 30-45 minutes before departure. So giving yourself two hours should be fine, assuming you’ve already checked in and have your boarding pass.
I know you asked for ideas other than the MOA. But this is the most commonly used and easiest location for passengers on long layovers. They have lockers of various sizes. Besides shopping, they have a movie theater, amusement park rides, an aquarium, miniature golf and more. If you want a table service restaurant that doesn’t feel like it’s in a mall, try Twin City Grill. It’s kind of dark and “clubby,” with a full bar.
The light rail transit is the cheapest way to get to MOA or downtown Minneapolis, but a taxi or uber might be easier with luggage. The fare is $2, but seniors can ride for $1 with the right kind of ID (check metro transit’s website). There are ticket machines at the light rail stations. Be sure to buy a ticket before boarding. You cannot buy a ticket onboard and, if caught without one, the fine is steep. Another place along the route that might be of interest would be Minnehaha Park. There is a popular seafood restaurant there (Sea Salt Eatery) and you can see Minnehaha Falls (of course, there’s still the problem of luggage). Take the light rail towards downtown and get off at the 50th Street/Minnehaha Park stop.
From the airport, you can get to downtown Saint Paul on Bus Route 54. This bus stops at the airport in the large parking lot next to the Silver parking ramp (and the rental car counter area). It seems to run about every 10 minutes on a weekday. The fare is the same as on the light rail.