Thanks for the advice, all!
I ended up throwing out one bag of rings, rinsed off the other and returned it to the filter, and added one new bag that I'd already purchased. I clipped the reused bag so I know next year which one is older.
I also replaced the old sponges. When I restarted the filter, wow, the tank got really cloudy in the space of 10 minutes. I guess a lot of gunk got released when I removed the old media. It's almost back to clear now, a couple hours later, but one thing I noticed was that the filtration process was likely slowed because when the tank is filled up to the rim, the water level in the three-chamber, built-in filtration system goes over the partitions between the filter column and the two pump columns, thereby allowing some of the incoming water to bypass the filter media.
I have a JBJ 28-gallon Nano Cube. Am I overfilling the tank? I fill it to the rim because not only does it look better that way, but also, if the level is even a little lower, then within a few days the water level dips to the point that the filter pumps get noisy because water isn't coming into the filter fast enough.
I'm expecting to hear replies to the effect that I should be adding water regularly between water changes, but I'm too lazy and forgetful to do that. Since the tank has been sustaining my fish pretty well for five years, perhaps what I've been doing is good enough, but am wondering what is optimal ...