You should use both options. Asking your deskmate politely to keep it down is fine. As is moving if possible. If all else about your job is ok, then you don't have any insurmountable issue to deal with. Just annoyances.
Your solution 3 is problematic.... sleep is important, without enough you will be easily irritated at noises and more sensitive to them. With plenty of sleep it should be easier to focus and tune out conversations, you can train yourself to over time.