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.
Always take it to Apple. Warranty or not they will fix broken screens for $100. You never know what type of LCD you're going to get on eBay or sham "we fix phone" shops.
door Justin Winchester