This is my very first plug-in so I'm going with trial & error, I have two fields a checkbox and a textbox, when I check the checkbox I have javascript that enable/disable the relative textbox.

What I do:

1. Check the checkbox -> The textbox is enabled by javascript
2. I enter a value in the textbox and save the settings — all good
3. I remove the check from the checkbox -> The textbox is disabled by javascript
4. At this point if I save, it strips the value of the textbox

Here are the fields definitions:

        add_settings_field(
            'stwc_ft_oosn_0', // id
            'Enable', // title
            array( $this, 'stwc_ft_oosn_0_callback' ), // callback
            'settings-admin', // page
            'stwc_ft_oosn_section' // section
        );

        add_settings_field(
            'stwc_ft_oosn_notice_1', // id
            'Text', // title
            array( $this, 'stwc_ft_oosn_notice_1_callback' ), // callback
            'settings-admin', // page
            'stwc_ft_oosn_section' // section
        );

While for the javascript it is disabling by jQuery adding disabled to the textbox, when disabled I can still see the text inside the textbox when I save the settings.