you can take advantage of the balanced input design of the 2882 to get more noise rejection than you thought possible on a guitar input.
In order to do this, you will need to make a psuedo-balanced telescoping shield guitar cable. This can be constructed with a TRS connector, a TS connector and balanced microphone cable. This cable will treat the guitar as a floating balanced source and provide a telescoping shield from the 2882 ground.
One of the most useful non-standard cables, but one rarely offered in ready-made form, is the pseudo-balanced cable. This can be used to help avoid or reduce ground loops when connecting unbalanced equipment to balanced inputs — for example, a synth output to a mixer input — the basic idea being to avoid any direct connection between the ground of the unbalanced source and the ground of the balanced input, while still conveying the signal.
If your computer audio device has unbalanced I/O, but you want to plug it into an analogue mixer or other device with balanced I/O, you can make up special pseudo-balanced leads that safely break the ground loop. Over the years I've soldered up lots of these to lower the noise floor of my external MIDI synths, and I've also used them to completely cure background noise problems when connecting the multiple audio channels of devices such as M Audio's Firewire 410 to my other gear.
You just buy twin-core screened (mic) cable instead of single-core screened, and then solder an unbalanced TS plug on one end (the soundcard end) and a balanced TRS jack one to the other (the mixer end). The 'tip' connections are made as normal using one cable core, while the 'sleeve' of the unbalanced end is connected to the 'ring' connection at the balanced end using the other cable core, and the 'sleeve' of the balanced end is either left unconnected, or preferably connected via a series eighth-Watt metal-oxide resistor of between 50(omega) and 500(omega) (I normally use 100(omega)) to the 'sleeve' connector at the unbalanced end. Such an approach will completely cure most soundcard ground-loop problems, and if you're going to solder up your own leads anyway it will only cost you a few pence more.
The diagram shows a TS (red and green) jack at one end. If you are
just re-soldering a TRS jack at one end (red, yellow, green in the picture) it will just mean that the ring (yellow) in that jack will not be connected to anything.
If you can't do it yourself, it shouldn't be too hard to find someone that can.
The TS cables you have are fine and should work. But you have a problem so this is a possible solution. You'll still need those cables, you'll just need to add an Ebtech HE-2 and 2 short TRS cables.
Start with the pseudo balanced lead and see if that helps, if not, consider the Ebtech. Laird is always right when he says to investigate one thing at a time. If you have a spare cable you can try swapping things around to check all the cables are fine. Sometimes even new cables can go bad.
There are more ways to use the ebtech, the way I described is just one I think covers the most issues with the least expense. You may also find a buffer helps (The lehle sunday driver is the most recommended).