A tweet by @BLAsia_Africa led me to a neglected passage in Eznik of Kolb, the 5th century Armenian writer, and a quotation from Paul the Persian! From it I learned that:
…the Avesta was transmitted orally and not written down!
The author drew this conclusion after reading some remarks by R. C. Zaehner in 1955:
However, whatever our view on the evidence of Paulus Persa, we have two other testimonies which can leave us in little doubt as to the fluidity of Zoroastrian dogma in Sassanian times. These are supplied by the Armenians Eznik of Kolb and Elise Vardapet. Eznik, like the nameless heretic of the Denkart, was struck by their inconsistency. ‘Their foolishness’, he says, ‘is enough to refute them from their own words which are mutually exclusive and self-contradictory’; and again, repeating the oft-made charge that they had no books, he says: ‘Since their laws are not in books, sometimes they say one thing with which they deceive, and sometimes another with which they seduce, the ignorant.’
 Ed. Venice, 1926, bk. ii, §2, p.128; Langlois, ii, p.375; Schmid, p.94.
 Venice, 1926, ii, 9, p.156; Langlois, ii, p.381; Schmid, pp. 111-12.
(Langlois = V. Langlois, Collection des historiens anciens et modernes de l’Arménie, 2 vols, 1867: p.179-251; Schmid = J.M. Schmid, Wardapet Eznik von Kolb: Wider die Sekten. Aus den Armenischen ubersetzt…, Vienna, 1900. Online here.)
There is actually a complete English translation, and I used to have a copy but it was mislaid. So let’s use Langlois, and just check the context of that quote. It appears in column 1 on p.381, in about the middle of the page:
En second lieu, pour cacher cette honteuse action, [Zoroastre] publie que pour le besoin des jugements [Ormizt et Arhmèn] ont créé [le soleil]. Aussi comme les dogmes religieux ne sont pas écrits, tantôt ils disent une chose, et se trompent, tantôt ils en disent une autre, et ils trompent les ignorants. Cependant si Ormizt était Dieu, il pouvait tirer les autres du néant, comme il avait créé les cieux et la terre, et non pas au moyen d’un commerce infame, ou bieu en raison de l’absence d’un juge.
Secondly, in order to conceal this shameful act, [Zoroaster] set forth that [Ormazd and Ahriman] created [the sun] to perform judgements. Also as the religious teachings are not written down, sometimes they say one thing, and are deceived, sometimes they say another about this, and deceive the ignorant. However if Ormazd was god, he could brings the others out from nothing, like he created the heavens and the earth, and not by means of an infamous commerce, or because there was no judge.
That does seem like a pretty clear statement that the Avesta, the Zoroastrian scriptures, did not exist in written form at this date as far as Eznik knew; and that in consequence Zoroastrian teaching was pretty fluid. I have seen popular claims that Christianity borrowed from Zoroastrian sources; but if there really are similarities, chronology would suggest that the borrowing is in the other direction.