There is nothing to be nervous for oath. It is a simple ceremony, with hardly anything which can be of a surprise.
Regarding proof of citizen, this is what it says - "In general, you must provide your U.S. birth certificate or U.S. passport. Other documents you may provide are a Consular Report of Birth, Certificate of Citizenship, or Certificate of Naturalization". So, you can go ahead with social security. Note that you will have to wait for a few days to get the info sync with SSA.