The core responsibilities of being a teaching assistant (TA) are
Leading weekly discussion sections
Holding weekly office hours
Grading exams
(Potentially) grading homework, or overseeing graders
The following are not responsibilities of a TA:
Substituting for the instructor in lecture
Writing exams
Working in excess of your employment level
Assigning course grades
Your job as TA is over on the date listed on your appointment letter. This is typically sometime after the final grades are due (usually the Tuesday following finals week). However with permission from the instructor of record you might be able to leave San Diego sooner. Be sure to talk to them about this in advance to see what they expect. Expectations at the end of the quarter typically involve scanning and grading exams and, if necessary, creating a final spreadsheet of grades. If, at the instructor’s discretion, you do leave town before the end of your employment period, make sure you are accessible. If there are last-minute problems, you need to be available to help fix it.