The vast majority my amazon apps are purchased, but not installed, however, I've purchased all of them via the phone app.
It's easy to check though - go to amazon.com, log into your account and scroll down to "Digital Content"
At the bottom of that list you'll see "Your apps and Devices"
Click that and you should see the app you "purchased" - as long as it's there you can install it from the apk on any phone you sync with that account (up to 10, I think) in perpetuity essentially. So it will be there when you get your new phone as long as you see it in your app list on amazon.