WP website won't accept any new content - need dedicated hosting?

I can't add any more content to the website. Theme devs said to increase limits on the server (memory_limit, max_input_vars, max_execution_time). We upgraded a package with the host, and tried to manually change these. However I can't do that because it's shared hosting.

Hosting support now suggested to upgrade our account to a cPanel, VPS or Dedicated Server, and move the site.

I never had to do that before, and I'm wondering if they're just trying to upsell, or is it actually necessary to get dedicated hosting? Any advice?

Thank you.