My best guess based on the current symptoms is that this problem is originating from your CMS. It seems like when your CMS encounters an emoji character, it drops the rest of the line.
One way you could verify this would be to insert the blockquote without the twitter-tweet class into your page i.e.
<blockquote><p>I am officially flipping my commitment from Florida to THE Ohio State University. Thanks for all your support. 🅾🙌 <a href="http://t.co/WxrvliX74Y">pic.twitter.com/WxrvliX74Y</a></p>— Baker Inc. (@Lastname_Baker) <a href="https://twitter.com/Lastname_Baker/status/524720617434533888">October 22, 2014</a></blockquote>
If that renders as…
I am officially flipping my commitment from Florida to THE Ohio State University. Thanks for all your support.
…then I think it’s safe to say the problem isn’t with the Twitter code.