I didn't see a setting in the plugin to set my timezone, but my WP timezone settings are correct. I'm UTC -7 and everything else appears correct.
The reason this is an issue is because all new tweet-posts are going into "Scheduled" rather than "Published". I tried adjusting the settings to push the new tweets to "Draft" (that worked), then back to "Published" (that didn't). When I open one of the scheduled tweet-posts in the editor, it shows that it's schedule to be published seven hours later (UTC time).
Also, I'm getting several duplicates popping up sporadically. And yes, I have RT's and @replies turned off. What gives?