The play was produced in 1727 by Lewis Theobald, who claimed to have somehow come by a lost manuscript of Shakespeare's late play Cardenio, based on an episode from Don Quixote. It's a little odd that there is no character called Cardenio in Double Falshood, the name of the character having been changed to Julio - by Shakespeare, or by Theobald? The manuscript itself has been long lost, believed destroyed in a fire in 1808. So a reasonable doubt has been hanging around the play since 1727.
Myself, though not an expert, I'm reasonably convinced that most of the first half is by Shakespeare - no particularly memorable quotes, but there's a feeling of the old master keeping his hand in. But I also suspect that Theobald edited it down - the play is much shorter, and the plot less convoluted, than we normally get with Shakespeare. A lot of the second half is clearly Theobald rather than Shakespeare or Fletcher, and the switch to eighteenth-century rather than seventeenth-century idiom is occasionally jarring.
To today's reader, the most disturbing aspect of the play is the rape of Violante by Henriquez, which takes place off stage between Act One and Act Two. Act Two then follows both Henriquez, full of guilty bluster, and Violante, injured and looking for escape, and it's in this very uncomfortable pair of scenes that one actually feels Shakespeare at work to convey the characters and feelings of two people, one of who has done something brutal and awful to the other. The rape is Shakespeare's invention; in the original Cervantes story, Dorothea is quite clear that she was seduced (and indeed married) by Fernando, who has deceived and abandoned her, but is not accused of assault. Today's readers will be squicked by the ending of Double Falshood, in which Henriquez is made to marry his victim Violante; they will be even more squicked by the eighteenth-century epilogue wondering what Violante was making such a fuss about.
I do wonder if this very uncomfortable theme was part of the reason that the play was lost. The First Folio includes several Shakespeare plays for which there is no contemporary record of performance, whereas it is known that Cardenio had several stage runs in 1614 and after; if Heminge and Condell had wanted to include it, they surely could have tracked it down. On the other hand a couple of the other late plays are also missing, so it may simply be that Heminge and Condell had better access to the earlier archives (or indeed that our records of missing plays are better for the later period).