This has been a much debated topic over the last few weeks.
According to Google Webmaster Tools regarding the disavow tool, "This is an advanced feature and should only be used with caution. If used incorrectly, this feature can potentially harm your site’s performance in Google’s search results. We recommend that you disavow backlinks only if you believe you have a considerable number of spammy, artificial, or low-quality links pointing to your site, and if you are confident that the links are causing issues for you. In most cases, Google can assess which links to trust without additional guidance, so most normal or typical sites will not need to use this tool."
So as you can see, most sites shouldn't need this tool. If you have thousands of spammy links coming from one source, with keyword heavy anchor text, that could look bad to Google; however, you are more or less averaging 1 link for 200 different domains. As long as that anchor text is unique (even if it doesn't completely make sense) you "should" be fine.
This article from Search Engine Land, 6 Things To Think About Before Disavowing Links, states "If you haven’t actually been penalized and you start disavowing your links, you’re essentially outing yourself to Google that you manipulated the system. Make sure that you equivocally know you were penalized and it’s not just some random fluctuation in rankings, a sitemap or indexing problem, or an accidentally no-indexed page."
You say, "Contacting these sites to have these links removed would be a very long process..." Google actually recommends you contact the site owners first and make an attempt at doing everything in your power to request your links get removed prior to using the disavow tool if you choose to do so.
Long story short. With that few links coming from that many unique domains, AND the fact that you haven't received a warning from Google... I would be tempted to just leave things and evaluate the keywords you were previously ranking for. If you see that it has been a straight up nose dive in rankings, there may be some concern, but fluctuation in rankings (at least for me) is relatively normal. In the course of a week, I can be high on page 2 or on page 5 (I have extremely competitive keywords) - I have come to expect this type of fluctuation, because I always rise to the top eventually.
Hope this helps.
Good luck.
Mike