My name is*****'m a certified computer professional with 15 years of experience. It would be my pleasure to assist you with this. I have to say that this sounds exceedingly strange. First of all, Facebook will never send you a request that you change your password. You may be asked to change it if you are trying to login and enter the wrong password ***** many times but they will never email you out of the blue and ask you to change it. Secondly, Facebook, does not provide direct phone support or live tech support. Therefore at the very least it sounds like you have been misled if they told you they were Facebook. Secondly, $230 is a LOT to pay for recovery of a Facebook account. I personally would not pay that. However, if you decide not to pay, you should have someone scan your computer for any software that they may have installed on your computer so they can't get back in and do any damage.
I have offered "Secure Remote Assistance" for you. This is completely optional, but I do recommend it. I will check for any remote access software that they may have installed and remove it, and in addition, I will scan your computer for viruses, spyware, and malware. When I am done, you can be sure that your computer will be secured and the "Tech Trager, Inc" will not be able to get back in.