There are three variants of the Greek text here:
(a) ... τῇ μεμνηστευμένῃ αὐτῷ γυναικί ... ("his betrothed wife")
(b) ... τῇ μεμνηστευμένῃ αὐτῷ ... ("his betrothed")
(c) ... τῇ γυναικί αυτου ... ("his wife")
Variant (a) is found in the majority of Greek manuscripts, including the Codex Alexandrinus (early 5th century). It is the reading chosen by the Textus Receptus, upon which the King James translation is based.
Variant (b) is supported principally by corrected versions of the Sinaiticus (4th c.), Vaticanus (4th c.), Ephraimi (5th c.), and Bezae (5th c.) manuscripts. It is the reading opted by the Nestle-Aland "Critical Text" edition, upon which most modern Protestant Bibles are largely based. It is also the reading selected in the Greek Orthodox Patriarchal Text.
The last variant, (c), is not found in any Greek text, but is implied in certain manuscripts of early Latin and Syriac translations.
The word μεμνηστευμένῃ is the perfect passive participle form of the verb μνηστεύω (mnēsteuō) which most lexicons translate as "betroth", but, to your point, also ambiguously define as "espouse".
It is not a common word. It appears only 3 times in the New Testament and only 8 times in the whole of the Greek Septuagint version of the Old Testament. According to the classical Greek references provided by Perseus, the word never meant any sense of "marry" or "be married", but related to courtship (seeking marriage). This can also be seen in the use of the word in Luke 1:27:
Εν δὲ τῷ μηνὶ τῷ ἕκτῳ ἀπεστάλη ὁ ἄγγελος Γαβριὴλ ὑπὸ τοῦ Θεοῦ εἰς πόλιν τῆς Γαλιλαίας, ᾗ ὄνομα Ναζαρέτ, πρὸς παρθένον μεμνηστευμένην ἀνδρί, ᾧ ὄνομα Ἰωσήφ
In the sixth month the angel Gabriel was sent from God to a city of Galilee named Nazareth, to a virgin betrothed to a man whose name was Joseph
"Betrothed" and married are not mutually exclusive here. As stated above, μεμνηστευμένῃ is a perfect participle; the meaning is that Mary had been betrothed at some point in the past. Since someone now married had once been betrothed, there is nothing strictly inconsistent by qualifying "wife" with "betrothed", even if it seems a bit redundant. I would suggest, though, that in choosing the words he did, Luke was sensitive to the fact that although Mary appeared to be a conventional wife to those who saw her and Joseph traveling together, their marriage had not been consummated as a "normal" marriage would have.
