First, let me say that appliancepartspros.com is AWESOME! The pricing, diagrams, videos, and ordering process are top-notch.
So, the problem didn't turn out to be either the drain pump or check valve, though the videos provided by APP made changing them out a breeze.
The problem was the drain pump cover. That part was not seated firmly and the small tab on the cover wasn't engaged in the sump wall.
The video provided by APP for the drain pump cover shows the technician using a flathead screwdriver to press down on the cover firmly and 'snap' it into place.
Now I have a new drain pump and check valve installed, the pump cover correctly installed, and we're washing, draining, and sanitizing our dishes once again!
I don't want my appliances to fail, but when they do, I'll be returning here before calling in a repair technician to the house!