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If the ebooks are from Amazon, and if they have DRM (encryption), then each ebook file is locked to a single device or app. What you describe sounds like DRMed ebooks from your Kindle EInk failing (as expected) on your Fire. One way to tell is to try to read them in Calibre, if this works then they are DRM-free.
You should be able to download Amazon Kindle ebooks for your Fire directly from Amazon. Amazon will give you a copy that is DRM-locked just to your particular Fire.
An alternative is to strip the DRM, which gives you a future-proof backup and allows a single ebook file to be managed by Calibre on multiple devices. See Apprentice Alf's Blog for more on DRM. Note that the 3rd party DRM-stripping plugins for Calibre are typically the easiest to use, but they work on import - so you would have to re-import your ebooks.
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