In order to thoroughly test your website for accessibility, all of the following types of testing should be regularly employed.
Having a tool like DubBot that checks your website for the most common accessibility issues is especially important when you have multiple content editors. Automated testing can catch issues early and actually train your content editors on what common accessibility issues are and how to fix them.
Test with a keyboard
Automated testing can do a lot of things, but it can't catch everything that keyboard-only users may run into when using your website. Use a keyboard to navigate around your website and evaluate the behavior. Is it easy to get around? Can you tell where you are on the page?
Test with a screen reader
Many users with a disability use a screen reader to understand a page's layout and content. You can use the native screen reader that comes with your computer, or purchase software especially for testing. In addition to making sure everything can be heard, also check for things that might annoying listeners, like links being read twice (both title and text), navigation read on every page that isn't skipped, etc.
Test with users
No testing will be more valuable than testing with the users who are actually affected by your accessibility improvements. Find a wide variety of users. In addition to those with no/low visibility and limited/no hearing, also including seniors and those with mobility issues.