According to one of John Mueller's latest hangouts it seems, unless you have a manual action against your site, you can stop worrying as don't have to actually remove links to your site if you've used the disavow tool. The disavow submission will, in effect, from Google's point of view, give the desired result. Of course, those links will still be on the net, so, if you really want to be squeaky clean, it still ought to be cleaned up.
A manual action will require those links to be disavowed, and removed.
The video starts at 43:20