Why is my bathroom sink leaking around the overflow seal?
I agree with Sfors on trying to seal up the leak areas, as this will make plunging a future clog in the primary drain possible.

Although, the rust and leaks being that high up tells me your overflow's outlet into the primary drain is blocked and the overflow is staying mostly full, instead of freely draining straight through.

The overflow's outlet is a side port hole just below the primary drain's opening, so you may have access to it to confirm it's a real drain (there was a very stupid period when they were just for expected appearance) and is cleared when fished from the overflow inlet above.

Now, if you don't see a side port inside the primary drain, then the wrong drain was put into the sink with the overflow being totally deleted as a drain and does indeed just fill up...until it finally turned itself back into a drain, the wrong way.