Your brain can store a finite amount of state because it's a finite piece of matter. You are therefore a finite automaton manipulating symbols outside of your brain, e.g. on a piece of paper. That makes you no more powerful than a Turing machine. In fact, since your lifetime is bounded, the amount of paper your can write on is bounded too. You are in thus less powerful than a Turing machine.

The fact that you can recognize some paradoxes means exactly nothing about your computational power. There is a sufficiently large lookup table that contains the correct answers to any question you might be able to read during your lifetime. Evaluating it requires nothing more than a finite automaton.