Well first off, you can't actually just replace the display glass on the newer iPhones. They've got a fused display, so you'll have to replace the whole thing. (unless you've got some specialized tools)
As for using a 6 screen on a 6s screen, I know for a fact there are dimensional differences, and the fact the new screens have the extra 3D Touch layer, which you'd be missing by using a 6 screen. I can pretty definitively say this wouldn't work well if at all.
So you've got a couple options,
you can wait for replacements to start being sold (actually there are some for sale on eBay but they are very expensive, like 400 bucks+) or you can go through Apple. In your case, this might actually currently be the cheaper option. You've still got your 1 year manufacturer warranty, so that would reduce the cost of the service right off the bat. If you picked up Apple Care you can use that as well.