Here is my advice: Unless you are ready to spend big bucks on full frame f/2.8 zoom lenses, stick with APS-C.
While a EF 24-105mm f/4 at f/4 on Full Frame will give you much better low light performance than an EF-S 18-135 f/4-5.6, it won't give you *that* much better performance than a EF-S 17-55mm f/2.8 at f/2.8 on APS-C. It will still be better, but IMO not throw all your EF-S lenses away and all the hassle you will need to do to upgrade better. You could use primes, but that does not sound like the flexibility you want.
If you really want to go full frame and want to use zooms, I would not do so until you can afford a 24-70mm f/2.8L II. Now that is a zoom that will really shine in low light and make the full frame transition worthwhile. The 24-105 f/4 on FF is an upgrade, but so is the EF-S 17-55mm f/2.8 on your current camera; I would explore the latter instead if you are not trying to break the bank. Once you can afford 6D + 24-70 f/2.8L II - then upgrade to full frame.
I don't think he's really "wasting" FF here though, he might not get a great deal of low light advantage from a 6D + 24-105mm vs a 17-55mm F/2.8 on his current camera BUT he will get improved range so its give and take.
One alternative that might be worth considering is the Sigma 24-105mm OS, reviews I'v seen tend to point towards it correcting the Canon 24-105mm's weaker performance wide open at the tele end.