How to manage data flow between UI components?

Is there a less convoluted way to allow data to flow between components?

Yes. It's called a model. In a model / view / controller pattern (MVC), an instance of the model is passed to the view so that the values in the model can be read by the view. The values in the model are updated by controller classes.

In a Java Swing application, you generally have many controller classes, each responsible for a portion of the view.

Take a look at my John Conway’s Game of Life in Java Swing article, to see how a fairly simple Java Swing application makes use of the MVC pattern.

What I normally do is have a JPanel that holds a reference to all the elements involved. This JPanel implements ActionListener so you can just pass "this" to every element (JTextField, JButton, etc.) that you create.

This way you have access to all the objects (they are instance variables) and listen to all the events within the same class.

If things get too messy then the ActionListener could be an external class (a "controller") that could hold a reference to more than one JPanel.

Some code:

public class TestJPanel implements ActionListener
{
    private JButton button;
    private JTextArea textArea;
    private JCheckBox checkBox;

    public TestJPanel() {
        button = new JButton("I'm a button");
        button.addActionListener(this);

        checkBox = new JCheckBox();
        checkBox.addActionListener(this);

        textArea = new JTextArea();
    }

    @Override
    public void actionPerformed(ActionEvent e) {
        if (e.getSource() == button)
        {
            textArea.setText("Button clicked");
        }
        else if (e.getSource() == checkBox)
        {
            textArea.setText("Checkbox clicked");
        }
    }
}