For example, I click a link and the appropriate event fires. Upon clicking a download link on the same page, the previous event persists. Specifically on an article page, I click the 'like' button that triggers an event. When clicking a download link within the article, the tag fires once with the previous event from the 'like' and then fires a second one that contains the appropriate event for the download link.
What are the conditions that trigger your "like" event. It sounds like on click of like your conditions evaluate to "true" and on click of a download link they also evaluate to "true," which is why the event is getting carried over. I would check your conditions and make sure they aren't overlapping.
Assuming that you are talking about SiteCatalyst, Tealium will automatically set linkTrackEvents for you which should keep events from carrying over into later server calls.
A page and steps to reproduce would be helpful.