Damped or heavy cases don't prevent fan and drive noise getting out because the case has holes in it, though they will prevent case vibrations.
I've been building very quiet PCs since the mid 1990s. The ine in my sig has two fans, never overheats and my decibel meter often can't tell it from room background noise at one metre.
I suggest you look at Noctua's website for a big cooler that fits your cpu.
After that cooling is all about getting enough airflow to prevent overheating while keeping noise down.
Then look for quiet fans, the bigger the better. Noctua again are pretty good. However you also need to be running them at low voltage to keep the speed, therefore noise, down. There are a variety of fan speed controllers on the market, or cables which convert the usual 12v fan power points to 5 or 7v.
Your next problem is the video card fan. These can be incredibly noisy. I use a fanless card, but there are editors about for gpu card's BIOS settings which allow you to adjust the pre-set fan speed vs temperature curve. Lots of cards fan the fan up to full speed as soon as the card is doing more than idle. By adjusting the fan speed settings you can delay fan acceleration to when the card temperature actually calls for more cooling.
Finally there's the power supply fan. Another source of noise. Seagate make good fanless power supplies so long as you watch the total power drain. A case that puts the psu at the bottom can help keep the psu cool. Heat rises, and a psu at the top of a case is usually sitting in the hottest spot.
Then you need to test everything to find the lowest possible amount of fan noise. I do this by loading the cpu to 100% and keeping it there (prime95 is a useful tool for that) while I keep an eye on cpu temperature. So long as it stabilises for 15 minutes while below the cpu's maximum thermal rating I'm happy. This makes the cpu a bit hotter than normal at idle, but really pays off sound-wise. It's very rare for a cpu to hit a sustained 100% capacity, so running closer to the chip's rated temperature doesn't seem to cause a problem.
I do a similar thing with the gpu. I also check drive temperatures, though that's not an issue with SSDs. Suspending an HDD so it's isolated from the case, or even putting it in the case bottom on a layer of foam can do wonders for quieting them.
It's also worth trying case fans in different locations and voltages. The PC before this one had a single case fan which I found gave greatest cooling at lowest rpm when placed at the back of the case (where air usually exits) blowing into the case, with the top front 5.25" bay left empty and open for hot air to get out.
www.silentpcreview.com is a useful resource, especially the forums.
http://www.quietpc.com are a UK supplier of quiet parts, as are
https://www.overclockers.co.uk. (no connection with either other than as a customer).