Recently my bathroom sink stopped draining. Thinking it might be clogged... I disassembled the drain and realized that it was not. While running the faucet at low pressure it seems to drain fine… however when running both hot and cold at full pressure it gets clogged. Could it be an airflow issue? What can I do?

So are you saying you opened the trap and it was clear? Rod straight into the wall but you will hit a 90 elbow so use a small 1/4" snake. Rod as far as possible, at least 6-8'

Also, and in addition to Ballenberg1's advice: you could remove pop up plug and see if there is hair ball on it.