It has been becoming increasingly evident that our old refrigerator which we got when we bought the house nearly 20 years ago is moving toward the end of its working lifetime. We might be able to get the current problem fixed, but putting money into a 20 year old refrigerator is not the best investment.
So -- after a lot of online research -- we went out to the appliance store today and bought a new Samsung refrigerator. The girls are (at least temporarily) happy, because it will have an ice and water dispenser. Yes, apparently it was time for us to move into the late 20th century on this. :)
We also decided to get a fridge with french doors and a bottom freezer. We think this will work out for us, but I suppose we'll find out...
Now we just have to make sure that the old fridge keeps running for another week.