Recourse for Non-refunded Flights?

Bought $2600 airline tickets for an international trip for wife and I's 20th anniversary back in October through a third party discount site that my wife gets through work. 4 weeks ago we were notified that the airline had changed our flight dates, which doesn't work for us.

Called the 3rd party to see about getting a refund, talked to someone in their foreign call center who said they would have to deal with the airline to get the refund. They call back a few hours later and say a refund will be issued in 2-3 weeks. 3 weeks was 2 days ago.

I called third party site again 2 days ago, got a different person who apparently had no record of the previous conversation re: getting a refund, and gave me the same line about needing to deal with the airline to secure a refund, and they would call back in a few hours. 36 hours later, haven't heard anything.

What recourse do I have in this scenario? Because meanwhile we had to book alternate tickets so we've now paid twice. Should I call my credit card company and try to get them to reverse the original charges from almost 4 months ago?