No need to pay for additional support, but you will have to make changes to your site.
The issue is that many of your assets aren’t loading (css/js).
While you haven’t changed anything, Netlify did change something, which perfectly explains it:
Since that deprecation you can no longer load a URL like:
https://rileyking.netlify.com/css/side-menu.css
Instead it must be:
https://rileyking.netlify.app/css/side-menu.css
This also explains the broken links:
https://rileyking.netlify.com/post/learning-from-failure-nitinol-fracture-mechanics-in-r/ (broken)
https://rileyking.netlify.app/post/learning-from-failure-nitinol-fracture-mechanics-in-r/ (works)
To fix you need to adjust your site so it no longer references netlify.com
You could do this by changing all of your links to Root relative
So instead of an absolute URL of:
https://rileyking.netlify.com/css/side-menu.css
You change the reference to:
/css/side-menu.css
If you cannot easily do that, then changing the .com to .app would suffice.