The text does not in fact state Joseph's age at death twice. The meaning of Genesis 50:22 is not Joseph's age at the time of his death, but his age at the time of the events in verses 24 and 25.
Verse 22 is a preface to verses 24 and 25 that indicates that the prophecy and oath related in those verses happened in the last year of Joseph's life, which is related in verse 26. The theological significance of this is that it says that the the brothers apparently did not know of the coming deliverance from Egypt until Joseph told them shortly before he died.
The linguistic support for this understanding is as follows.
The verb form ויחי, y'hi, "he lived", in verse 22 indicates a span of living or a continuation of life but does not imply a cessation of living afterwards as does the English translation "he lived". For example, Genesis 5:3:
וַיְחִי אָדָם שְׁלֹשִׁים וּמְאַת שָׁנָה וַיּוֹלֶד בִּדְמוּתוֹ כְּצַלְמוֹ וַיִּקְרָא אֶת שְׁמוֹ שֵׁת
in which the NIV translates ויחי as (in bold):
When Adam had lived 130 years, he had a son in his own likeness, in his own image; and he named him Seth.
That is Adam lived 130 years, and then Seth is born. But then Adam then continues to live until the age of 930 years, as related in Genesis 5:5. We could also translate this "When Adam was 130 years old".
Likewise, Genesis 5:6
וַיְחִי שֵׁת חָמֵשׁ שָׁנִים וּמְאַת שָׁנָה וַיּוֹלֶד אֶת אֱנוֹשׁ
The NIV translates ויחי with "when" to indicate continued living:
When Seth had lived 105 years, he became the father of Enosh.
Likewise Genesis 5:9 (MT/NIV):
וַיְחִי אֱנוֹשׁ תִּשְׁעִים שָׁנָה וַיּוֹלֶד אֶת קֵינָן
When Enosh had lived 90 years, he became the father of Kenan
and similarly in Genesis 5:10, 5:12, 5:13, 5:15, 5:18, 5:19, 5:21, 5:25, 5:28, 11:16, 11:18, 11:20, 11:24, 11:26.
Compare also with Genesis 47:28 (MT/NIV):
וַיְחִי יַעֲקֹב בְּאֶרֶץ מִצְרַיִם שְׁבַע עֶשְׂרֵה שָׁנָה וַיְהִי יְמֵי יַעֲקֹב שְׁנֵי חַיָּיו שֶׁבַע שָׁנִים וְאַרְבָּעִים וּמְאַת שָׁנָה
Jacob lived in Egypt seventeen years, and the years of his life were a hundred and forty-seven
Here, the same verb, יחי, is used to indicate the span of time that Jacob lived in Egypt, seventeen years, which was not his age at death.1
The verb ויחי is also used at the end of a phrase to indicate continuation of life as in Deuteronomy 4:33 (MT/NIV):
הֲשָׁמַע עָם קוֹל אֱלֹהִים מְדַבֵּר מִתּוֹךְ הָאֵשׁ כַּאֲשֶׁר שָׁמַעְתָּ אַתָּה וַיֶּחִי
Has any other people heard the voice of God speaking out of fire, as you have, and lived?
And here it is obvious that the meaning is to continue to live, as it is also in the following verses:
Deuteronomy 5:26 (MT 5:22):
כִּי מִי כָל בָּשָׂר אֲשֶׁר שָׁמַע קוֹל אֱלֹהִים חַיִּים מְדַבֵּר מִתּוֹךְ הָאֵשׁ כָּמֹנוּ וַיֶּחִי
For what mortal has ever heard the voice of the living God speaking out of fire, as we have, and survived?
I Kings 17:22 (MT/NIV):
וַיִּשְׁמַע יְהוָה בְּקוֹל אֵלִיָּהוּ וַתָּשָׁב נֶפֶשׁ הַיֶּלֶד עַל קִרְבּוֹ וַיֶּחִי
The LORD heard Elijah's cry, and the boy's life returned to him, and he lived.
In fact, there is no verse in the OT that uses the verb ויחי, and he lived, that also indicates the death of the subject. The fact of death, or the total number of years of life is only ever mentioned in subsequent verses. Where the fact of death is not mentioned in a subsequent verse. The reader can only infer it.
In Genesis 50:22, יחי is the verb of the second clause that is parallel to ישב, yeshev, dwelt, in the first clause. Both verbs indicate present continuing action, and indeed, after the death of Joseph, the other brothers continue to dwell in Egypt.
Genesis 50:26 then is the only verse that indicates that Joseph died when he was 110 years old, in the same year as the preceding prophecy of and oath related in verses 24 and 25. Without verse 26, we could conclude that Joseph lived longer than 110 years because the events of verses 24-25 are subsequent to the statement of age in verse 22.
1. In fact, the text is not absolutely clear about the actual age of Jacob when he died. We must surmise or infer it.