Can I perform Shashthi Shradh if the exact death date of my ancestor is unknown?

If you do not know whether your ancestor died on Shashthi tithi specifically, you should not perform Shashthi Shradh for them on this day. Instead, you can perform their Shradh on Sarva Pitru Amavasya (the last day of Pitrupaksha), which covers ancestors of all tithis. However, if you have an approximate Gregorian date of death, a learned pandit can look up the corresponding tithi in the Hindu Panchang and confirm the correct day for you.