PSU Mostly Nuked my HDD (Mostly), Advice on getting data back

So my computer went down after I heard a short. I reboot and my computer takes ages to boot once I get to the desktop.

It never fully finishes loading the programs in the background but I have mouse movement. So I shutdown and take a look around in my case. I smell the smell of burnt electronics coming from my PSU. I decide not to try a reboot until I put another PSU in there.

Fast forward, I get a new PSU and I put windows 7 x64 on another HDD and am able to read from the affected HDD. It reads super slow and I'm able to get some files off of it, but then the file transfers time out and then the drive just disconnects from the system. If I do a hard reboot, I can access the drive again, but it again times out every time I try to move files.

I'm pissed at myself because I was going to build a second rig to act as backup as well as a general work computer, but my drive failed. I need the files off that drive though as it has important files on it.

I'm not quite sure what to do, I've seen an ebay service that can replace the board and flash the right firmware on said board for the drive, but not sure if it will fix the issue. What can I do?

Also, it seems one other drive has a similar issue. It shows up in BIOS, but not in windows at all. Although this drive isn't nearly as important.

Have you tried your affected HDD on different SATA ports/a SATA to USB adapter?

I have not, I don't have a USB to Sata adapter that can run a 3.5" Drive. I'll try a different port though.