After reviewing the other repair stories related to this "7 blinking light" syndrome, I contacted their support staff via Chat. I relayed my symptoms along with model number of dishwasher. She indicated that this would normally mean an issue with either the heating element, the high limit thermostat, or the control board. Since the control board was so expensive and (luckily) out of stock, I ordered only the heating element and the thermostat.
After pulling out the dishwasher and removing the existing heating element, I tested it with an ohm meter. It was still testing okay. I was getting the same reading on the old element as the new one that I received. But, since I had the dishwasher all apart, I decided to go ahead and replace the heating element anyway. The dishwasher was over 13 years old, so I figured it was time to replace it.
I also ordered the high-limit thermostat since it was relatively inexpensive. Removing the clip that holds this thermostat was a bit more work than I expected. It really took quite a bit of pressure to get that clip dislodged from the panel holding it in place. I finally used a flathead screwdriver to give the extra push on the clip that was needed.
We have not experienced the blinking clean light since these repairs were completed. Since I replaced both items, I can't say for sure whether it was the heating element or the thermostat. But, for $100 with shipping, it was a relatively quick and cheap way to get our dishwasher working again.