Just to clarify above and add a little bit.
The Disavow file to Google only goes to Google. Even when you submit it there, and Google confirms that it has the new copy, it takes a while (several months) for Google to then account for the domains you are Disavowing. i.e. Don't expect an immediate effect from Google in what you see in Search Console and/or in the SERPs. Also, side note, you should generally disavow at the domain level as once you have a bad link from a site, the whole site is probably not worth it and it makes your job easier if there are a bunch of links from the same domain. Also, spammers will use a subdomains to create copies of the main site and add even more crappy links to your site. Disavowing the domain takes care of the subdomain as well. The exception to this would be sites that are on wordpress, as all the subdomains would really be from different sites (you get the idea) so you would disavow at the subdomain level for those sites.
I have one site I run that gets a bunch of spam links to us built and so 1x to 2x a year I have to pull everything I can get from Moz, Majestic, Search Console, etc and find new domains to add to the Disavow file. It sucks, but it helps even though Google's official line that they can account for negative SEO.
There are tools such as CognitiveSEO that will aggregate links for you from various sources and then you can upload your Disavow file there and it will then know to filter out all the links it finds from those domains to help you weed out what you need to audit and I have found this useful.
Continuing from previous answers, Moz has no idea what the disavow file you sent to Google contains and will not take that into account when it is looking for links. You may see more links from the disavowed domains show up in Moz because of that. This is where tools like CognitiveSEO are helpful in being able to sort out things. I would hope that for MozPro members this would be a feature to add so that your OSE profile could account for Disavowed links. I have found Moz helpful in using the Spam score to decide if I should disavow domains for Google.
Regarding your point about finding links to your sites and then the sites no longer exist. That happens. The Moz link database and also Google's databases are not updated in real time. I have also seen that the links in Search Console are not up to date with what I see in Google search. If you go to the sites and they are 404/410 Gone, you are probably fine. I would say the disavow is a good conservative way to make sure that you are "safe" and would recommend that you move forward with that. If the sites are gone (as mentioned above) Moz should show them gone in the next update or two (depending on if they crawled those sites or not).
Bottom line on all this is, are you seeing less organic traffic to your site? All of the above may be a moo point if the bad sites that link to you are dead and potentially even Google is already ignoring them (and the disavow will make sure of that). Google does not look at the Moz metrics and so it does not matter if the links are included in the Moz index as long as you know they are taken care of. If your organic traffic is ok, then enjoy that coffee and relax as you submit the disavow file.
Cheers!