Do not, I repeat do NOT use the disavow tool. If you have backlinks out there that are broken, use 301 redirects to pages with similar content. If you use the disavow tool, I promise your domain's visibility will be a whopping 0.
Bad advice all around but thanks for trying to help. Webmaster Tools is a pitiful place to check links, in most circumstances they display between 10%-30% of links. When they do end up showing links most of the time they are links that have been removed for MONTHS. Heck, I even have links showing up that I had removed over a year ago. Use backlink checkers such as ahrefs to look at links, not Webmaster Tools (although it should be one source for links but not THE source for links). And don't use the disavow tool, that's just a wasted effort.