I'm thinking I need to check the value of each of those fields each time any one of them is updated but that seems kind of burdensome
That is the easiest way to do it.
The trick is to create a simple API so that your solution scales whether you have 2 text fields or 20 text fields to check. In other words you don't do the check based on variable names.
You can create a class to register text field you want to check and store the text fields in a List. You add a DocumentListener to each text field. Then you simply iterate through the List whenever a DocumentEvent is generated.