I would think any fine spray lubricant would work. I've never used any graphite lube on cables. The first thing to do is figure out if things are gummed up.
The problem you have with integrated brake/shifters is that you have to replace the entire unit, not just the shifters. For what you do, I would probably replace with an 8 speed Alivio/Acera brake/shifter, an M410 looks decent. An M410 for the rear derailleur. I usually buy off ebay and the cost isn't too bad.
As already mentioned, you may want to replace cables and housings and a chain, KMC is cheap, but that is up to you.
John