I figured out how to solve this problem but still do not understand why it is happening. The problem is that the phones were connecting to the mobile network even when wifi was available. This can be avoided by turning off the "Mobile Data Usage" option on the connections/data usage page. This then requires that this option be turned back on when you leave the wifi coverage area so it's kind of a pain.
This doesn't explain, however, why this suddenly started happening. The usage patterns of the phones in question had been the same for years before we encountered this problem. These two phones were in different locations and therefore using different wifi networks. Both of them started this behavior at the same time so it seems like something must have changed in the software of the phones themselves.
In the meantime, I spent $60 in data overage charges before figuring out how to solve the problem.
The messages I got from Verizon could have explained this most likely source of the increased data usage but they did not. Instead, they just offered to sell me plans with more data.