This subject is discussed by many meforshim, particularly within Chassidut.
The first name pertains to the final redemption (which is why it is being utilized to address Moshe in the first redemption) and also precedes creation (meaning before the Kav enters the Makom Panoy and what follows afterward).
Regarding your observation that in Shemot 3:12 this name appears to be used as a verb is not quite correct. It is actually referring to two dimensions or aspects of the same name, similar to the usage of the name Havayah twice at the beginning of the 13 middot of Rachamim (In this case with Havayah, the two uses of the same name are referring to two different aspects of the same name.). The first Eyeh is the Asher aspect of the second Eyeh in the complete name.
Havayah, the name which was known indirectly to the Avot through its being 'dressed' in the name שדי, pertains to each and every generation. This is one of the inner intentions we have when reciting the first line of the Shema. That the word Shema itself (שמע) emphasizes this (The ש״מ of שמע is שדי-יהוה meaning 340).
This is what is meant in the conclusion of Shemot 3:15.
וַיֹּ֤אמֶר אֱלֹהִים֙ אֶל־מֹשֶׁ֔ה אֶֽהְיֶ֖ה אֲשֶׁ֣ר אֶֽהְיֶ֑ה וַיֹּ֗אמֶר כֹּ֤ה תֹאמַר֙ לִבְנֵ֣י יִשְׂרָאֵ֔ל אֶֽהְיֶ֖ה שְׁלָחַ֥נִי אֲלֵיכֶֽם׃
טו וַיֹּאמֶר֩ ע֨וֹד אֱלֹהִ֜ים אֶל־מֹשֶׁ֗ה כֹּֽה־תֹאמַר֮ אֶל־בְּנֵ֣י יִשְׂרָאֵל֒ יְהוָ֞ה אֱלֹהֵ֣י אֲבֹתֵיכֶ֗ם אֱלֹהֵ֨י אַבְרָהָ֜ם אֱלֹהֵ֥י יִצְחָ֛ק וֵאלֹהֵ֥י יַעֲקֹ֖ב שְׁלָחַ֣נִי אֲלֵיכֶ֑ם זֶה־שְּׁמִ֣י לְעֹלָ֔ם וְזֶ֥ה זִכְרִ֖י לְדֹ֥ר דֹּֽר׃
The two names mentioned in 3:14 and 3:15 pertain to different concepts.
Just as a point of reference, the quotation from Or HaChaim mentioned by Joel K is referring to a related, but different name affiliated with the creation and how it alludes to a particular permutation of the name mentioned in Shemot 3:14.
That reference, אל עליון, which is also mentioned in the opening blessings of the Amidah prayer, is referring to G-d's name צלמו which has the same gematria (166) as עליון. This is the name of G-d that we learn from Bereshit 1:26-27. Three of the letters from that name (צלם) were used in the building of Adam's wife, Chava. Those three letters together with the kollel (a gematria of 161) allude to the name referred to by Or HaChaim, namely the expansion of Eyeh with Yuds like this (א״לף, ה״י, י״וד, ה״י).