It is never an easy task. But you can begin by deleting (from the list you have downloaded) all the links that you don't feel there is a problem with, and those which you know have already been removed but are still showing in GWT. In my case, from an initial list of around 4000 I was left with about 80 I wanted to look at individually (further investigation showed that many of them were no longer present, even though still reported by GWT).
If you have multiple links from the same domain, you can attach a list of the all urls where they are located (copied from your GWT download) and attach them to one email request.
It follows that the more links you have and the greater the number of linking domains, the greater will be the work load in doing this.