Tests: Fix `--headless` flag

[n7studios]

Summary

Use --headless=new in acceptance tests, as required by Chrome 109+. Resolves issues with some tests failing.

Testing

• Existing tests pass.

Checklist

• [x] I have written a test and included it in this PR
• [x] I have run all tests and they pass
• [x] The code passes when running the PHP CodeSniffer
• [x] Code meets WordPress Coding Standards for PHP, HTML, CSS and JS
• [x] Security and Sanitization requirements have been followed
• [x] I have assigned a reviewer or two to review this PR (if you're not sure who to assign, we can do this step for you)