New HDD on USB3.0 won't stay connected, either.

This new USB3.0 ext HDD seems to be in an almost constant state of Windows thinking it is unplugged, then just plugged back in, back and forth. I had to shut autoplay on it off because it would happen at least once a minute, sometimes 30 seconds, sometimes a couple minutes. The drive and case are brand new, but this is just like what was happening with my old drive via eSATA. I thought something was just wrong with the drive or my eSATA port, but now this. I know the USB3.0 port is fine because I have plugged in flash drives (even a 3.0 one) and connected other drives via 2.0 in it, with no problem. As for the eSATA ext HDD at least, this was happening a couple months ago before I wiped my system, and is still happening now. New OS install changed nothing.

These drives show up as known and installed in Device Manager. Disk Management's info is correct. Acronis is the same. Sometimes they just need a coaxing from "Scan for Hardware Changes" to even try to let me do anything. Windows Explorer, though, shows it all on its own, when it does think it's plugged in. (& yes, the plug fits quite snugly into the port, all is fine there.) Also, on my fiance's rig, both drives are also behaving exactly as the should. It's something over here...

Any ideas what could be causing my system to have trouble keeping these external drives onboard?

try this.
To Disable Selective Suspend:

open the control panel
go to power options (you may have to set view to small icons)
click change plan settings
in the window that opens:
click change plan settings
click change advanced power settings, expand usb and ensure usb selective suspend is disabled.

It doesn't solve many problems, but it does solve some connectivity issues with usb devices.

Ok, I checked that. It was already disabled, though. =/

Sorry that was the only one I really had for you.
Trying multiple cables, and checking the usb driver would be all I have for you outside of that.
If the drive has a optional external power supply you may want to try using it as well.