Honestly, I don't know how you would research this. However, a quick search of Wikipedia
indicates that indeed, you are a citizen by nature of having been born to one, and thus your children would be by the same law. How you go about claiming your citizenship though I don't know. I think you may need to contact the embassy to ask about it.
We are facing a similar thing one day, as I am Canadian and my husband is American. We live in Canada. But its very easy to get information about applying for documentation for American citizens born abroad.