Winter in Ohio never got too warm for me, either!
You can make the cabinet work, though. But rather than mechanical like nails, screws, I like glue and clamping with only a few fasteners. Fewer holes to patch and hide on the outside. There are some really good wood glues that cure pretty fast and will hold as much as the wood fibers do. In your case, I might look at some of the construction adhesives in a tube that are used for winter work. Just don't use too much so that the clamping makes it run out all over?
I like using a full length piece of wood like a 1X2 for adding a top if the top is thick enough to hold a screw well. With 1X glued to the sides, top put on and glued, I add just one or two screws for confidence that it will stay forever. Up from the bottom to be out of sight. The 1X will help add strength to the sides also. Sometimes that top rim board looks like a cat might break it! Sometimes the old houses I've remodeled were so crooked that cabinets would never fit the walls but there are lots of ways to cheat them to looking right and holding up long term.