There is Google's Disavow Links tool, but it can harm your site if you're not really qualified to use it. Here is the tool where you can read all about it:
It is better that you work with the other webmasters and persuade them to remove the links. If you explain nicely that you may have to use the disavow links tool, that should motivate them to help you.
It's a shame to give up an aged domain, assuming that it is. You essentially have to start over. Can you delete the pages on the site with the adult content on it? If so, I would do that and clean up any links that might break as a result.
Perhaps you could secure the new domain right away, but only use it if/when you have to. It's better to fix the old site than to start over, given the damage is reparable.
And regarding the search engine damage, you'd have to determine if you've been labeled a "bad neighborhood" site. Here's an article that explains what that means: https://blog.kissmetrics.com/seo-street-smarts/
However, it may not be too late and if you act quickly, you may be able to get this cleaned up.