Just a minor point really, but one thing which really stood out for me the first time I sat down to go through the work done on them was just how difficult the idea of 'a bible' is to pin down. I suppose that was a legacy of the JW belief of there being one fixed divine word which has remained unchanged down the ages. And their claim the Dead Sea Scrolls prove that because all the OT is there unchanged. Well that ain't so.
Instead what we see in the Dead Sea Scrolls is a group of texts which are passed down with few changes, often none. So the Torah, Isaiah, the Psalms and the twelve minor prophets seem to be fairly unchanging. But then everything else seems up for debate. It's not fixed. It's altered and amended and revised. They add new texts and new stories and new prophetic visions. They even rewrite books to form new books. A new version of the old process we see in what we call the canon with Deuteronomy and Chronicles.
And one does see the same process within christianity too. Oral traditions get converted into written ones, and not only does the perspective change with the writer but also what actually is said to have happened. It's hard to escape the majority accepted conclusion that the gospels are re-writes from the same source material (probably Mark). Newly discovered letters are merged with older ones (eg the known pseudographical letters which made the canon vs those accepted to have been written by Paul), and then selection from amongst the various visions to get Revelation into the canon. And before the canon process semi-finalised things, what a crazy mix of texts were being used on a regular basis. eg The Shepherd of Hermas.
Similar processes take place in Christianity as those we see happening with the Dead Sea Scrolls. The OP mentions the shared 'pesher' interpretation of prophesy, and the general atmosphere, but I think one sees a really strong Judaic strand to 'scripture' in its entirety too. And it's very, very different to the idea of a fixed word of God which passes unchanged through the ages.
(References: Florentino Garcia Martinez is an interesting read on the scrolls generally, but his contribution to Authoritative Scriptures in Ancient Judaism (2010) supports the bulk of the factual assertions I've made here. He kindly shares a large number of his essays and lectures on academia.edu if anyone has interest in learning at that level of detail).