I work in a university doing research and many of my personal friends are regularly publishing academic authors. From my observation I can say that those writing in a highly formal style talk like they write in private, too. It is their language. They read almost exclusively formal writing, they converse in it about sophisticated topics, and they use it when they speak with their children (thereby adding to the advantage that children from an academic background have over students from non-academic families), and when they write they don't have to switch styles or make an effort to maintain it.
If you want to master formal writing, you must make formal language a habit, both in input and output.
On the other hand, much of academic writing today is written in a more informal style, which in my opinion often eases comprehension and increases my reading pleasure. The demands of your tutors and professors aside, you should not worry too much about the formality of your language, but rather about its understandability and clarity and, beyond that, feel free to develop your own style. This may take some time, a lot of reading, and much writing and trying out what works best (for you).
As for first or third person in academic writing, there are a few questions and useful answers on this site. Use the site search to find them.
In the past it was the norm to write in an impersonal style to give an impression of objectivity. Today objectivity must be proven in the description of your methodology, and every action must clearly be attributed to its agent by the use of personal pronouns. Because of that, academic texts today are generally written in the first person singular or plural, depending on the number of authors.
The APA Manual explains this in much detail and with many examples, and I recommend you to read it even if you format your text following another style guide.
Finally, you really shouldn't guess what your teaching assistant meant, but ask him or her to explain their critique and give an example in your text.