"I was so drunk last night. I left my table and didn't say anything to anyone."
More natural to shorten "did not".
Probably better to join the sentences as well:
I was so drunk last night that I left my table and didn't say anything to anyone
I left the table quietly, and went to my bed and slept till morning.
"left" is better than "skipped". "Skipped" doesn't sound right since it implies a bit of "energy" (which from the text doesn't work since you've said you have a bit of a hangover. 'Skip" can mean "miss", but you would be misusing the word if that's what you meant as you are already at the table.