Okay, your code is stripped from that payload and it never come to the HFC plugin to process provided settings.
Here is payload I’m getting (request URL is …wp-admin/options.php, your is okay – options.php 302 POST method)
option_page=head_footer_code_settings&action=update&_wpnonce=f47ef275e4&_wp_http_referer=%2Fwp-admin%2Ftools.php%3Fpage%3Dhead-footer-code&auhfc_settings_sitewide%5Bhead%5D=%3Cscript+id%3D%22warmly-script-loader%22+src%3D%22https%3A%2F%2Fopps-widget.getwarmly.com%2Fwarmly.js%3FclientId%3De3e5df6e3e5e34bc5ad47319b22b2d61%22+defer%3E%3C%2Fscript%3E&auhfc_settings_sitewide%5Bpriority_h%5D=10&auhfc_settings_sitewide%5Bdo_shortcode_h%5D=n&auhfc_settings_sitewide%5Bbody%5D=&auhfc_settings_sitewide%5Bpriority_b%5D=10&auhfc_settings_sitewide%5Bdo_shortcode_b%5D=n&auhfc_settings_sitewide%5Bfooter%5D=&auhfc_settings_sitewide%5Bpriority_f%5D=10&auhfc_settings_sitewide%5Bdo_shortcode_f%5D=n&auhfc_settings_homepage%5Bhead%5D=&auhfc_settings_homepage%5Bbody%5D=&auhfc_settings_homepage%5Bfooter%5D=&auhfc_settings_homepage%5Bbehavior%5D=append&auhfc_settings_homepage%5Bpaged%5D=yes&submit=Save+Changes
Can you try in incognito (it could be that some browser extension mess with payload). If it works in incognito, what browser extensions do you use?