What to do when old links to dead pages still showing on google searches?

Solution: 301 redirect them to the most relevant existing pages.

The 301 redirect is sending users to a different page. The best case is you have similar pages on the new site you can direct users too. If you don’t, you can redirect to the homepage.

301 redirect them to a page that's related to the old page, if there is nothing you can point them to serve a 410 to let Google know the page was removed.

You should also update your sitemap and submit to Google Search Console and Bing Webmaster tool.

Also, Next time you do a redesign make 301 redirects a part of your planning process. :)