How does the break cable go through the shifter?

Shift interlock. Possible brake switch.

check the Shifter cables they break. You have to remove center console surround to make sure cables are still attached. Very common problem of eyelets breaking. The shifter should feel sloppy. If shifter feels tight, the transmission may be bad

Maybe shifter cable came off of the shifter inside the console ! Sometimes plastic break that hold the cable to shifter lever or down at the transmission or maybe the shifter cable broke !

... do you, by any chance, have non humbucking pick ups on your guitar? Just curious...

I'd revisit the adjustments first before spending that kind of money. It sounds like cables have stretched and simply need to be readjusted, which may be normal.

http://www.parktool.com/blog/repair-help/categories/derailleur-systems

Ask yourself this... Why would the left shifter have anything to do with this and how do you know you'd get an 'improved' set on a new purchase?

Maybe look on eBay periodically for a deal or a stray right shifter.

Aren't boutique parts fun?

There is a main fuse that is blown or it could be a fusible link. Anyways there is a hole near the shifter and you can stick something in there so you can move the shifter manually. Sometimes it is a key slot sometimes it is just a hole. That hole is there so that if an event you don't have power to release the shifter you can do so manually.

Billettechnologies.com sells a replacement cog to put your shifter backinto action. Their part replaces a factory plastic piece...no wonderthese are failing at a fast rate.
Here is the LINK to make this change out of parts...

This really depends on what cable you need to replace, the brake or the shifter.

For Shifter:
1. Shift the all the way down to the lowest gear using the small inside shift level for both the front or rear.
2.Disconnect the cable from the derailluer, likely a 5mm allen wrench will be needed.
3.Pull housing away from the shifter exposing the cable where it enters the shifter.
4.Compress the brake lever, this will expose a small hole on the inside of the shifter where the cable head is seated. For the right shifter the hole will be on the right, for the left shifter the hole will be on the left.
5. Push the cable in from where the cable entered the shifter. The head of the cable will now be exposed.
6.Pull the cable the rest of the way out of the shifter.
7.Install the new cable.

For Brake:

1. Disconnect the brake cable, likely with a 5 mm allen wrench.
2. Compress the brake lever. Keep it compressed until step 5.
3.Look straight in from the front of the bike, you will see the head of the brake cable.
4.Push the cable out of the shifter from the exposed area of the cable, then pull once you can reach the end of the cable.
5.Install a new cable.

Depending on the reason you have removed the cable, it is advised that you also replace the housing in most cases. For the shift cables this is not a problem, however, for the brake cables you will have to untape the bars to access the housing.

I hope this anwsers your question. Thank you