I don`t think I would invest in a cooler for such an old card, especially one that begins to show signs of failure.
Saphire and Club3d make passive versions of the hd7750 which should be a decent upgrade over the 9600gt. There`s also the Powercolor 7850 though I would avoid such a powerful passive card. You could end up needing to increase the speed of your case fans thus defeating the point of a passively cooled card.
If you`d rather have an nvidia card, your should look into a gtx650/660 with an aftermarket cooler.
Thanks for the replies everyone - they are very helpful.
The ECS 9600GT came with an Accelero cooler attached to it (S2, I think). My case is an Antec P280, roomy with a few fans running undervolted at low speed. GPU temps have always been well under 70C, and I blow out the dust regularly.
The artifacts occur during boot up and randomly in windows, so I don't think it is a heat issue. Graphics drivers were updated when the artifacts began to appear, and that made no difference. My hardware is otherwise reasonably recent (Intel i7-2600, etc) so all i need is a functioning graphics card that meets or exceeds the performance level of my current card.
I've read the more recent reviews of various cards here at SPCR that do mention that fan cooling for graphics cards has vastly improved, but since i don't need a raging game monster, I'll stick with passive - one or two less moving parts to fail, and graphics card fans have been a chronic problem in my experience.
Looking around, the selection of passively cooled cards is disappointing. If I needed something more powerful, I'd go to the trouble and expense of a higher end card and add an after market cooler, but for my uses I don't think it is necessary.
In addition to the cards you mentioned, Asus makes a 7750 passively cooled card. The card and cooler combine to be pretty enormous, but there is plenty of space in my case, so something like that should work.
Are there issues going from Nvidia to AMD on an existing Windows install? I've always used Nvidia cards so never had to switch drivers before.
Just looked up the card that the OP has, the ECS comes prefitted with the Accelero S2! I remember this card now, there were a few 9600GT's that came with it. The S2 has the heat capacity for the 9600GT but needs a little airflow for some cases. I tried both the S2 and then the S1 and the S1 is a big upgrade.
In light of this, the heatsink should be just fine. It would be worth stripping the card down, cleaning and rebuilding just to make sure that the cooler is on OK.
If the card itself is definitely on the way out then I would suggest taking the heatsink off and reusing it on a new card. Although I can't give any guarantees of mounting hole compatibility, the S2 should be able to work with any modern low/mid range card up to around 95W. The GTS650 would be a good guess.
Given the temps I'm seeing I am almost certain the cooling is not an issue. The RAM sinks are all still attached, so i don't think it is overheating RAM that is the trouble.
I do have an 8600Gt in another machine that has a fan cooler on it and the fan has begin to whine a bit more than it did when new. I hadn't thought to pull the Accelero off the 9600 for future use - excellent suggestion.