yes contacting FB sounds to me the best thing you can do, and never ever give your password away no more to other people.
Basically there are couple of steps to unlock the account, first of all, a security check to make sure you are human. Then, Facebook ask you to change your email password(s) for security reason. Here you need to change your password. Finally, Facebook shows some useful account safety tips for you. Thatz it and you are done. I think with over 500 million users, Facebook takes account security very seriously.
First off..... Never give your password/codes to anyone for what ever reason... and if you do... make sure that you change the password/code as soon as you can. Secondly... You can click on the forgot password/code button and facebook will send you an email to your email account linked to the fb account... or you will need to contact fb help/admin to have them help you.