Your travel agent is supposed to pass your CC info to the cruise line and they are supposed to charge you directly. If you ever have a travel agent that charges your card instead of having the travel provider charge it, that's a huge red flag. All that is just to say that yes, it is indeed Princess who charged your card and who now has your money.
It is also true that once you book through a travel agent, you can't call the cruise line directly and speak to someone there. The booking belongs to the TA and they are the only ones that the cruise line will speak with. I'm surprised that they were willing to discuss their cancellation policy with you at all.
So what should you do now? I'd go ahead and file the dispute with your credit card. Make sure you have a copy of the policy that was in effect at the time you cancelled and be prepared to send that in. You may not win this dispute though, especially if Princess has not cancelled the cruise and they're following their current policy for refunds, so you might end up having to wait out the 60 days anyway.
The risk of filing a dispute on a charge that's already been promised for refund is that you can get a double refund as the CC company refunds and then the cruise line refunds and the notifications going back and forth end up crossing. If that happens, the CC company can freeze your card while they untangle things on their end, and it will take them a lot longer than you will think it should. Also, some companies refuse all future business from people who file CC disputes. I don't know if Carnival/Princess is one of them, but I have definitely heard of rental car companies that will put you on a "do not rent" list if you dispute what they consider to be valid charges. A potential ban on future cruising might not be that big a negative for you, so consider it or not as you wish.
Thanks for you excellent explanation.
I have used the "wayback machine" to see and print a copy of the cancellation policy the day before I cancelled, while it does not mention CC refund time limits, it does say their FCC (which I'm not concerned about too much) will show up in 10 days. Of course it has not.
If I get a double refund, of course I'll notify the CC and Princess to take back the correct amount, so that does not worry me too much as I have lots of credit cards I can use, if one is frozen.
Being banned by Princess, would suck if the entire CCL company banned me, but we could still live with the other choices for cruise lines.
I do imagine, since the dispute is not the entire refund, it will complicate things a bit and I'll need to watch the dollar amounts carefully.
AirBnb which must also be getting slammed with cancellations processed my refund in 2 days.
So I've decided, to dispute the charge, as Princess has processed my cancellation (it shows to Princess rep as cancelled) 16 days ago, and they are holding onto my money far beyond normal processing times.
I'll wait until next week, in case my refund shows up in the next few days.