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Yes, St. Thomas actually accuses him of holding a Nestorian error! But he had not read the apologies against the Iconoclasts - my understanding is the De Fide Orthodoxa is all that had been translated at the time into Latin. The problem is solved when you realize that for Damascene, like the other Greek Fathers, procession, or "from the Son", is almost always used of procession from a first source or origin, whereas Latin theology uses the terms in a much looser sense. Properly the Spirit proceeds principally from the Father and consubstantially from the Son, as was explained to St. Maximos when he inquired:

With regard to the first matter, they (the Romans) have produced the unanimous evidence of the Latin Fathers, and also of Cyril of Alexandria, from the study he made of the gospel of St. John. On the basis of these texts, they have shown that they have not made the Son the cause of the Spirit--they know in fact that the Father is the only cause of the Son and the Spirit, the one by begetting and the other by procession--but that they have manifested the procession through him and have thus shown the unity and identity of the essence. (Letter to Marinus)

I had this exchange with Agrarian (LINK)

[Agrarian quoting me:] "If the Orthodox understanding is that the Son and the Holy Ghost proceed symmetrically from the Father, and not sequentially, first the Son and then the Holy Ghost, then indeed that is not compatible."
[Agrarian responding:] I don't recall the Fathers using words like symmetrical, but if I am understanding you correctly, then you are on the right track to understanding what Orthodoxy teaches.
One father wrote (can't remember which) that each person has a characteristic unique to Him: The Father is the source, the Son is begotten, and the Spirit proceeds. Orthodox arguments regarding the filioque are ultimately patristic, scriptural, spiritual, and practical, and not based on theoretical or metaphysical arguments of symmetry or asymmetry, but the observation was made by a father, and I thought I'd pass it on.
[...]

Note that Agrarian repeats what is also the Catholic dogma, misstated by me with the loose usage of "proceed": that the Son is begotten and the Spirit proceeds, but the implication is that there is no co-procession from the Son. I wish he could respond to your post here.

I am a visual person, so let me make this picture:

Catholic Trinity

Father	-- > eternally begets -- >	Son

\|
V
proceeds
\|
V
Holy Ghost

Orthodox Trinity

Father

eternally begets

proceeds

Son

Holy Ghost