Although... now I think about it... when I'm anxious (I have reasonably mild but fairly broadly scoped anxiety), I can become more emotional/instinctive/reactive to things, and my ability to reason can be greatly impacted as well.
So I do think there is some indirect correlation, in practice.
Emotional intelligence does not imply that you are driven by emotions. It imply that you understand them, which makes it easier to control how you react.
Someone who is emotional is not displaying emotional intelligence at that moment.
Often when I feel certain ways, I don't have reference points to mentally articulate how I'm feeling - if I even have the ability to consciously distinguish the feeling and highlight it. That shuts down a lot of internal dialog and analysis before it has the chance to take place.
When I'm anxious, my thinking is clouded across the board, which makes this weakness all the more apparent.
I wonder if there are any books out there that specifically help to instil an understanding of the nuances in emotional processing.
