My Kindle 3 Touch device has broken, and I'm getting the new Paperwhite version.
Will the cover case I've already got for the Touch fit with the Paperwhite, or are the device designs different enough that that won't work?
As far as I know, there is no Kindle 3 Touch. Kindle 3 readers have a keyboard and are therefore larger. Their covers will be too large for a Paperwhite.
If you are talking about a Kindle Touch (4th generation): There are non-original covers that claim to fit for both, so your old cover should fit. On mobileread they do however claim that the Paperwhite is slightly smaller:
The PW is 4.6" x 6.7".
The K4 (non-touch) is 4.5" x 6.5".
The K Touch is 4.7" x 6.8".
An advantage of buying a new (original) Paperwhite cover is that It has magnets inside. These will automatically switch your Kindle on and off if you open or close the cover.