The OP hasn't been back since December 11th, 2021 so I doubt we'll ever know what happened in their situation.
Welcome to TUG.
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Hello, OP here. You are completely correct to assume that I wouldn't return. It's usually the case, however I've had in mind to come back and give an update on what happened precisely because I know how frustrating it is to come across a post from someone who dealt with the same problem and to find that they never said what the resolution was. I'm very grateful that forums like this one exist and there are regulars around who are willing to try to help out complete strangers even though they know the strangers will most likely disappear without a trace.
Initially, following the intense feelings of shame and anger at having been so silly as to fall for this scam, all I could do was wait and see what happened. After looking at the pros and cons, and taking into account our anger, we made the decision to cancel the credit card. We had paid the initial deposit in cash, so the card had no debt. We just weren't willing to watch FACV take our money each month and decided that it seemed like the risk of debt collectors and the like was very low. We had three months from the data of signing before they were going to actually make the first charge, so we waited a month or so before making this decision.
In the meantime, I had actually also written a complaint to PROFECO. I think it was the same day that I made the initial post here however it might have been the following day. In this letter I detailed the exact dates of our stay and the details of the sale, including the name of the salesman. I had noticed that someone other than us had initialed the pages of the contract for us and I highlighted that this was the case as well. (If the forum elders here don't see a problem with it, I'd be happy to share what I wrote in a slightly redacted version.)
Anyway, this complaint was sent somewhere around December 11. On January 14, I received a notification that it had been received by them. On February 10, they informed me that they would be meeting with representatives of Grupo Posada on the 14th. On February 28th they informed me that Grupo Posada had agreed to refund the deposit and cancel the contract. In this email they requested bank account details where the refund should be made.
Following this, it was another few weeks before someone contacted us from Fiesta Americana, where they again asked for our bank account details. They then also requested that we send a signed letter saying that we give up the right to do a chargeback on this payment. I have no idea if this is legally binding in any way but after reading through it we decided that we didn't mind signing that letter. The refund (minus the CC charges of ~$250) was then made a few weeks after that.
So in the end it was an extremely lucky outcome for us, thanks to PROFECO, and thanks to me sitting down and wasting most of a day writing the complaint instead of joining my wife and friends at the Dallas Zoo. I'm not sure if it made any difference, but I did write the complaint in Spanish. All the subsequent PROFECO correspondence was in Spanish as well, so it seems like it would be hard to go through the process if you don't understand it to quite a high level.
Finally, although we had cancelled the credit card, they agreed to cancel the contract before the first payment was actually due, so we don't know what they would have said/done once they realized.
If I can help anyone out who's in the same situation, I'm happy to try. I wasn't getting notifications from my account here, so I'll try to turn those on.
I appreciate any help! Thank you
