airfare.com airfare dot com airfare.com did not ticket my reservation and lied to me, Internet

Here is what happened. I found good prices on airline tickets shown on airfare.com. I tried to buy them, providing my credit card information. However, I just wasted my time and the tickets got more expensive, while airfare.com claimed I have a reservation but did not issue any tickets to me.

Multiple attempts to have a meaningful conversation with them failed, because no one in their call center (obviously located in India) spoke good English, and understanding each other was almost impossible.

They gave me multiple contradicting reasons about why the tickets are not issued, ranging from "Virgin Atlantic does not accept Discover Card" to "we have to issue the airlines a check, since they do not accept any credit cards, and you have to pay us 40 dollars to issue the check." Well, none of this happened, so requested cancellation of my ticket, and did not hear from them. Called again and they hung up the phone just saying "Goodbye, have a nice day".

Now I cannot buy tickets from another website since I am not sure if airfare.com is going to cancel my reservation or issue the tickets. And even if they cancel it, I am not sure if they will charge me or not, and even if they don't charge me, the ticket prices have gone significantly up during the 2 days I wasted to deal with them.

What airline doesn't accept payment by credit card these days? Whom are they fooling?