We have had the same experience with a highly rated foreign airline on two occasions over the past month.
Their web site would lead us to believe that there are not many seats left. When we were trying to find the best combination of days and hours for our overseas trip, on each try the price went up until it was a thousand dollars higher and the choice of seats showed only middle of the row available.
On both occasions we telephoned the airline. The recorded announcement said it would cost us $20 more for a ticket and that there may be better deals on their web site.
Better deals? In both cases the so called nearly sold out plane turned out to have ample desirable seats and at the normal price!
So what is going on here? Are their web site computers programmed to measure interest and drive up price while shrinking choice?
In both cases it was well worth spending the extra $20 to speak to a a reservation clerk.
I’ve noticed that too! It may be that blocks of seats researched for third party sites are not released until a few days before the flight.