Do we have tools to check our questions and answers for broken links? I regularly stumble across them. We should fix them, either with changed URLs or with links to the Internet archive, to preserve the integrity of the answer, a good reader experience and (when only accessible by the Wayback machine) the content itself.
I am not aware of any such tools, as useful as they would be.
On Meta SE I found Is it possible to automatically mark broken links? which sounds similar to what you are asking for, so I recommend that you vote for it (like I just did).
In the meantime I think that we can only "fix 'em as we find 'em" as per How should we deal with questions that have broken links?
Upvoted the question because it would be useful if we could find tools that would check for broken links.
As I hear of websites undergoing redesign, I try to go back and re-edit my older answers to put the new links in, or to link to the Wayback Machine as needed.
One of the things I would also like to see cleaned up are the questions and answers that have naked links (with no labels), or with labels that are opaque (i.e. "this page"). If the link label is a searchable phrase, a user can look for the label on Google whether the underlying URL has human-readable elements in it or not.
Update: as I write this, the Evidence Explained website is undergoing redesign. I need to edit all my answers (and there are many) that link to EE's Quicklessons. The EE page is referring people to the Wayback Machine's copies of the old site in the meantime.
The Archive of Quicklessons is available via the Wayback Machine here. https://web.archive.org/web/20150905101409/https://www.evidenceexplained.com/tags/quicklesson
I am willing to use a local Markdown editor to make the changes if no online solution exists.