While שְׁכִינָה doesn't occur in the תנ״ך, the verb of its root שָׁכַן does, as well as שֵׁ֫כֶן … n.m. dwelling [physical structure]; שָׁכֵן … adj. inhabitant, neighbor; מִשְׁכָּן … dwelling-place, ‘tabernacle’. שְׁכִינָה is ususally associated with כְּבוֹד־יְהוָה֙. The one instance in the תנ״ך that כְּבוֹד־יְהוָה֙ is the subject of a verb form of שָׁכַן is Exodus 24:16 (וַיִּשְׁכֹּ֤ן כְּבוֹד־יְהוָה֙) in the context of Moses receiving the Law at Sinai. שָׁכַן is indirectly associated with כְּבוֹד־יְהוָה֙ in Exodus 40:35 (שָׁכַ֥ן עָלָ֖יו הֶעָנָ֑ן וּכְב֣וֹד יְהוָ֔ה מָלֵ֖א אֶת־הַמִּשְׁכָּֽן) context of the Tabernacle, with parallel clauses. In 1 Kings 8:11–12 כְּבוֹד־יְהוָה֙ and שָׁכֵן are in adjacent verses, but apparently have no close association (context of the Temple in Jerusalem).
Thus, the original meaning of שְׁכִינָה appears to relate to Moses and the Law. While שְׁכִינָה has been interpreted as associated with כְּבוֹד־יְהוָה֙ elsewhere in the תנ״ך, the Glory of the LORD כְּבוֹד־יְהוָה֙ is usually with verbs meaning filled or appeared.
Given the context, שְׁכִינָה appears to be feminine because it is an abstract noun meaning dwelling. In other words the meaning is not a concrete object such as a dwelling place or person dwelling. It is characteristic of an action of the כְּבוֹד־יְהוָה֙.
Abstract nouns [non-concrete things] are [often] morphologically feminine. Van der Merwe, C., Naudé, J., Kroeze, J., Van der Merwe, C., Naudé, J., & Kroeze, J. (1999). A Biblical Hebrew Reference Grammar (electronic ed., p. 179). Sheffield: Sheffield Academic Press.
I would like to get responses from those better at Hebrew, especially the Hebrew use of this word.