Thank you for all your replies, but I think this may turn out to be a fool's errand.
The offending machine is the Inspiron 530s: http://www.reusetek.com/dell-inspiron-530s-computer-desktop/
I Googled "Inspiron 530s" and "noisy fan" - there seem to be many threads on the Dell community about my very complaint. Solutions include upgrading the BIOS (I gutted the VIsta install for linux mint and I have no idea how to do it with a linux install) to replacing the fan, which worked for some, not for others, to reapplying thermal paste, etc.
I guess this is just an inefficient design. The fan is just too noisy! Those complaining weren't even using it for audio, just ordinary computing.
First, note that there is a Dell 530 AND a 530S model. Yours is the slimline "S" model. When searching fans for the Dell 530S, Google brought me to a bunch of 530 complaints about the fan. Not the same computer. Be sure that you are discriminating between the two in your reading. Also note that if the fan is the volume problem, sure. Just switching out the fan might not work. You have to know how to buy a quieter one (lol). There are clueless people on the Internet that try things without understand what they need to do
Still, you should open the case and see for yourself. Make sure it is unplugged before starting. Here's the disassembly information on your case and components: http://support.feelpcs.com/pc-spec/Inspiron530S/parts.htm. Once you get the cover off, information about the chassis fan is near the bottom of that page.
1) Plug the computer back in and listen to the computer with the fan still connected.
2) Turn off the computer, unplug the chassis fan, and turn the computer back on. Is the computer a lot quieter?
If so, then replacing the fan should help. No doubt that Dell uses cheap fans in those units, not ones that people would run in quiet PCs. Best I can guess from what I could quickly find, the Dell 530S chassis fan is an 80mm (find out how to measure the size yourself). For $10 to $20, there are definitely quiet 80mm fans. Since that is your primary case fan and it is smaller, best to not cheap out in that range, but go with one that has a higher CFM rating.
And don't forget to clean the fan blades when you first open the case. At least three or four times over the years, I have encountered dirty fan blades that were causing the fan to run unbalanced and make more noise. And as someone else pointed out to you in this thread, if your computer is 7 years old and the inside has never been dusted out, it could be running hotter, which means the fans might be running faster (motherboards often control fan speed based on temp).
