The Power of the Most High as the Holy Spirit
I think anyone accustomed to the Hebraic parallelisms of the Bible will recognize in Luke 1:35 an appositive parallelism whereby the Power of the Most High spoken of is made identical to the Holy Spirit:
Luke 1:35 And answering the angel said to her, The Holy Spirit shall come upon thee, and the Power of the Most High shall overshadow thee: on account of which also the Holy One to be born of thee shall be called the Son of God.
Note here that at once we learn the reason Christ is to be recognized as not a mere man, or 'mere' son of Mary, but who He is in truth: the Son of God (cf. Jn 1:1, 14—Lk 1:43). Gabriel is saying 'on account of this conception,a nd by the power of God and not by the involvement of man, you will concieve' (the thrust of Mt 1:25)—in answer, mind you, to Mary's inquisitive and mild objection, "How is this to be [accomplished], seeing as I know not man?"—a euphamism for 'do not have sexual relations' and so 'wonder how I am to concieve.' It was her will to remain a virgin, (cf. 1 Cor 7:34,37) as is in keeping with the "mother of the Lord," and "the Holy One;" "God among us" (Mt 1:23) and "the Lord of Glory," (1 Cor 2:8), even "the First and the Last" (Rev 1:17-18)! cf. Ezek 43:1-2; 44:1-2.
So the the overshadowing being the answer to the question 'by what means shall my concieving a son be, if not by intercourse?' shows us the function and purpose of her being overshadowed: to work the miracle of the Incarnation (Mt 1:18). No matter the referent of the overshadowing (her soul or her body) it is necessary that she be prepared in both to contain the Word of God (Wis 1:4).
Mary's 'Fullness of Grace'
The traditional rendering of the Greek word κεχαριτομένη (her-having-been-graced) as 'full of grace' is certainly a valid one (especially since it stands as a vocative 'title,' and in place of her name), and in any case bears witness to the Chirstian people's belief concerning its significance. It probably best renders the sense of the word, which denotes a completedness to God's 'gracing' her (much like how 'burned' denotes the completedness nature of the act of burning, and now 'having been burned' characterizes the thing which was burned.)
Significantly, the only other time this verb χαριτόω (to grace/show grace) is used in the New Testament is in speaking of the action by which God "[redeemed]" Christians, and made them "holy and blameless" before Him: 1
Ephesians 1:3-7 Blessed be the God and Father of our Lord Jesus Christ, who has blessed us with every spiritual blessing in heaven by Christ: just as he chose us in him before the foundation of the world, to [make] us holy and blameless before him, forordaining us in love unto the adoption as sons, (through Christ) unto himself: according as it shall have pleased him; [all] unto the praise of his glorious grace, with which he graced us [ἐχαρίτωσεν] in the Beloved, [and] by which we have redemption, through his blood—the forgiveness of sins—according to the plenitude of his grace.
We might say that at the very least, Mary was sanctified even from her mother's womb, or "from the first instance of her conception" (Pius XII, Munificentissimus Deus) in the same or in a greater way than was John the Baptist from his mother's womb (Lk 1:15): him a herald of the Lord, whereas she is His mother and the one who clothed the Him with His very flesh (Lk 1:43; Jn 1:1,14; Gal 4:4). As the Ark was made according to specific measurements and to be covered with pure gold and carried reverently on poles gilded with gold, so we expect Mary to have been chosen and prepared in a much greater way: who didn't contain mere symbols of Christ (manna which fell from heaven in the desert, the rod of Aaron, the tablets of stone with the Debarim or Words of God inscribed on them: Heb 9:4) but Christ Himself (the living Bread, born in the city called House-of-Bread, which comes from heaven Jn 6:48-51, the true high priest Heb 4:14, and the Word of God in the flesh Jn 1:14).
It's noteworthy that Mary, the Mother of God, is nowhere said to herself have been graced or filled with the Holy Spirit in preparation for her unparalleled role of honor, except being described as 'already' "having been graced;" if anything, she could be seen as carrying the Holy Spirit with her due to whom she was carrying in her womb, meaning the 'overshadowing' was during the whole time "the Word...tabernacled" (Jn 1:14) in her:
Luke 1:41-44 And it came to pass that when Elizabeth heard the salutation of Mary, the infant leaped in her womb; and Elizabeth was filled with the Holy Spirit. 42 And she cried out with a loud voice, and said, Blessed art thou among women, and blessed is the fruit of thy womb. 43 And why is this given to me, that the mother of my Lord should come to me? 44 For behold as soon as the voice of thy salutation sounded in my ears, the infant in my womb leaped for joy.
Inspiration vs. Overshadowing of the Holy Spirit / The Significance and Usage of 'Overshadow' in the Bible
Inspiration of the Holy Spirit is nowhere correlated with being overshadowed by the Holy Spirit. ἐπισκιάζω (to overshadow; envelope; [h/c]over over by presence) is a Greek verb used relatively few times in the Bible, and every time (with one exception) it refers to the direct presence of, or being surrounded by, God's glory.2
Interestingly, in concordance with the connections between the Ark of the Covenent and Mary discussed above, and elsewhere, God's glory cloud also filled or 'overshadowed' (Ex 40:35 LXX) the placed where the Ark was kept in the Tabernacle (Ex 40). cf. Rev 11:19-12:1,5.
1 In the Greek Old Testament we have another use in Sir 18:17: "Behold, is not a word better than a gift? Even these both are with a man graced [by God.]" (Vul. justificáto—made just/justified).
2 That I can find: Mt 17:5/Mk 9:7/Lk 9:34 (God speaks from a cloud which 'overshadows' them at the Transfiguration of Jesus); LXX: Ex 40:35; Ps 90:4; Ps 139:8 (God's most intimate presence or protection). Heb 9:5 uses a synonymous verb meaning roughly the same thing, describing the cherbubim of glory 'positioned as they were above' (κατασκιάζοντα) the mercy seat of the Ark of the Covenant. In Prov 18:11: "to the rich man, his wealth is the city of [his] strength, the great glory of which surrounds him."