I can imagine the shock, I've found the same situation with responses, I have a supposedly 1st cousin that isn't technically possible based on my tree as I know it. However they didn't return my e-mails, nor have several others, so it may not just be your situation, just a lot of people don't seem to reply back sometimes.
I'm helping out someone who is in the same situation as you, and it's been an interesting learning curve for me on many levels. If you have a possible "lead" that is always great.
What I would suggest to start is to upload your DNA to Gedmatch, and do this for both you and your sister. I believe also family tree DNA allows free uploads of your ancestry file to look for matches there as well.
This will help provide a "bigger" pool to compare, and Gedmatch offers some interesting tools, where you can run a report of all who match you, then run a report of all who match your sister and then remove the ones that match and that will leave you with hopefully a solid group.
If you think you have solid leads I would also suggest using the last names you think might be it and look to the addresses near where your parents lived at that time, and even see if those names might be on the backs of old family photos from that time frame. Sometimes a DNA mystery can be explained by looking at the neighbors on a census :)
Some of the gedmatch files have trees attached to them, so you can look for the names in common, but I have also found that those on gedmatch tend to be better at replying back asking to view trees, as they have all gone through the extra step of uploading your information there and sharing their tree with you.
I wish you luck, and hopefully answers soon.. And just a few do you have another family member to compare with? EG do you know for certain you have different dads? Or is it possible different moms? And have you ruled out the possibility that a younger or unwed sibling got pregnant and the child raised by your parents?
I think you will find out who the family is, and even if not "the exact person" likely find your family :)