In general, if you are not an electronics service engineer, there are two approaches.
Strip it apart & bathe it liberally in contact cleaner.
Take it to a service engineer.
1 covers dust & dirt, 2 covers component failure.
Your headphones should be safe plugged into the monitor sockets but you're unlikely to hear much. The monitor sockets are intended to feed an amplifier or amplified speakers with a medium input impedance. Α 10 kΩ load is suggested. It's not particularly critical but 32 Ω is too low. You'll need to add a headphone amplifier between the monitor outputs and ...