Sorry about your situation OP and I went through something similar with Merge/Carbon in 2010, which was finally resolved in 2014.
http://forumserver.twoplustwo.com/sh...postcount=5561
http://forumserver.twoplustwo.com/sh...postcount=6138
I too played on the same computer at some point as someone else had, that ended up owing money. The main difference in your case seems to be, they are giving you the option to pay the other players debt off. I agree it sucks but if you're expecting to find some way to get your account reinstated without the debt being paid, good luck with that. I think the only reason I was finally able to get my account reinstated after around four years, is because my relatives account was on another poker skin that's no longer on the Merge network. If not for that, I presume I would still be banned as well.
If it were me and it once was, I would try to find out how much is owed, then decide if it's worth it to you to pay it off to be able to play on Carbon. The way I looked at it at the time was, even though frustrating, the fault was mine for not being more careful about accessing my account and the potential for it being linked to the account of others. Yeah it sucks but from the pokersite's POV they have to do what's best to protect themselves.
As a matter of fact I've had this situation with Bovada/Bodog and Pokerstars as well, with Bovada it was a very low amount owed which they eventually waived because I told them the relative was deceased but with PokerStars it was a larger sum and I ended up paying it off to get my account reopened.
(FYI this was all back when family, friends and I first began playing online poker around 2008, being ignorant to the potential consequences of playing on the same computers, locations, etc). The point is, in my experience it's not unusual to be required to pay the debt of another player account if linked to being accessed from the same computer(s).