What a paradox feels like
I've previously mentioned how convenient it is that paradoxes don't affect us the same way they do a more formal system. Gödel, in his day, managed to prove that all complex formal systems have unanswerable paradoxes built in. That threw the mathematicians of his time for a loop, but nobody seems too concerned about paradoxes in the brain. This statement is false. See? You're fine.
But you could easily make the argument that those kind of paradoxes don't really affect our thinking. The same way that I can type the letters "this sentence is false" into a computer without causing any problems, we can think about an idea without believing in it. But there are contradictions that do appear in our beliefs and how we make decisions. I'm a particular fan of the trolley problem series of dilemmas, which often reveal contradictory ideas you've held for a long time without realising.
However, it's rare to see a trolley problem seriously affect someone. In most cases I think people do a good job of resolving contradictions, usually by not believing one of the beliefs anymore. In fact, it seems like most inconsistencies only last until you realise there's an inconsistency. But, to me, it's not a paradox if you can just solve it; with a paradox there's meant to be no way out. I think what would qualify as a mental paradox is a situation where you believe two contradictory things and you're not able to stop believing in either of them.
When you believe in something in a way that you can't stop believing, that would look like reality. So we're looking for a situation where the reality is impossible. Where you have to take option A, but you can't, so you have to take option B, but you can't. This is the sort of situation where you're completely, utterly trapped, and there are no options. I think what you feel in that situation is simple: despair.
In which case, perhaps despair is nothing more than what a paradox feels like. On the one hand, that means paradoxes can be very harmful. But on the other hand, perhaps knowing that you're experiencing a paradox could provide some comfort, as well as a healthy dose of encouragement to check your assumptions. Actual paradoxes are, after all, quite rare – Gödel notwithstanding.