Yes, we have some limited archaeological evidence to back up linguistic evidence. This area of research is pretty recent and hot, but the consensus seems to be that "rough copies" were prepared on clay tablets, and then transferred over to "fair [final] copies" on parchment or papyrus. Ironically, it was the rough drafts (on clay) that were preserved for us in the fires which destroyed the Bronze Age centers, while the nice copies were destroyed. That's the theory, anyway. There is a great deal of linguistic evidence to support the practice, nicely summarized by a series of writings by the venerable Gregory Nagy, and conveniently online via the Center for Hellenic Studies at Harvard over here.

For the Minoans, we have some pretty good evidence that they preserved long-form records on parchment (which, note, is specially-prepared leather, not necessarily paper, though papyrus was also around). They seem to have "closed" these rolls with seals which both secured them and identified their contents. We have some 700 of these seals extant from at least nine Minoan sites on Crete. That's a healthy amount. A similar situation is posited for Linear A and Linear B (Mycenaean).

Linguistically, there is an interesting theory (mostly by Palaima) which notes the odd conservatism of the Linear B script, which seems more suited for "painting" mechanics (like writing or painting on parchment) than for a more incision-style on clay tablets. He notes that the conservatism of Linear B's cursive elements over time probably points to the existence of a written medium beyond clay tablets. It's an interesting observation. For that, see Palaima, T. G. 2003. “‘Archives’ and ‘Scribes’ and Information Hierarchy in Mycenaean Greek Linear B Records.” In Brosius, M., ed. 2003. Ancient Archives and Archival Traditions: Concepts of Record-Keeping in the Ancient World (Oxford, 2003,)153–194.