If Wikipedia can be trusted, the oldest surviving manuscript is the Florence Shahnameh, dating from 1217 CE, "200 years after the final completion of the epic poem in 1010". This manuscript is not illustrated. Also according to Wikipedia, the Florence Shahnameh is "is incomplete and contains only half of the Shahnameh."
Since Wikipedia can be wrong, it is worthwhile to check other sources.
According to an end note in The World of Persian Literary Humanism by Hamid Dabashi, a specific story (described in a section of the book that I cannot access)
dates back to the earliest manuscripts of Ferdowsi's Shahnameh, such as the famous Florence manuscript of 1217, in which there are two prose introductions, in one of which the story appears.
According to The Shahnameh: The Persian Epic as World Literature, also by Hamid Dabashi, the scholar Djala Khaleghi-Motlagh, whom Dabashi describes as "the eminent Shahnameh scholar of our time",
the oldest manuscripts of the Shahnameh (...) are from the thirteenth century, from which we have two copies in Florence (1217) and London (1276). From the fourteenth century are nine copies now in libraries in Istanbul (1330, 1371), Leningrad (1333), Dublin (1340, another undated), Cairo (1341, 1394, a third undated), and Mumbai (undated).
I could not find information about the completeness of the Florence Shahnameh in Dabashi's The Shahnameh: The Persian Epic as World Literature, but at least Dabashi confirms that this manuscript is the oldest that has survived.
It appears to be difficult to establish what the oldest "complete" manuscript is, since scholars of critical editions of the epic try to get as close as possible to what Ferdowsi might have written by comparing several manuscripts. Some of these manuscripts are treated as base texts, while others are treated as secondary.
The "Moscow edition" (1966-1971), edited by Bertel et al, used the London/British Museum manuscript (1276) and the Leningrad manuscript (1333) as base texts; at that time, the older Florence manuscript, discovered in 1977, had not yet come to light. The more recent edited by Djalal Khaleghi-Motlagh et al (1990-2008) used the Florence manuscript and the British Museum manuscript as its base texts, treating the Leningrad manuscript as secondary. (See Dick Davis, Review of The Shahnameh, vol. 3, ed. Djalal Khaleghi-Motlagh, Mazda Publishers in association with Bibliotheca Persica, 1992, International Journal of Middle East Studies, Volume 27, Issue 3, August 1995, pp. 393-395).