Switching it to the other PCI-e slot didn't solve the problem.
The fan was fine.
Updated drivers failed to fix it.
So a new card seemed to be the solution. Given that it was probably time to go for a more capable card than the Radeon 1950 Pro I got last time, but didn't want to have to change anything else, an ASUS HD4850 1GB seemed to fit the bill. It's just 9" long, which means I can get it into the case (I have one free slot in the drive cage behind it, into which the tip of the card protrudes - anything longer than 9" is out as a result.) It uses only one 6-pin additional power connector, so doesn't need a PSU upgrade.
My 3DMark score just doubled, which considering the mobo didn't change is not bad. And I'm now wandering round Northrend with all graphic settings set to max, with a decent framerate, going ohh, ahh at the view. I can now see the real difference between the 'classic' landscapes and the new Lych King ones.