OED does place the etymology of "giddy" squarely as one possessed by a god.
Old English gidig insane, is shown by its guttural initial to be a graphic variant of gydig < prehistoric gudīgo- , apparently < Old Germanic gudom god n. and >int. The primary sense thus appears to be ‘possessed by a god, ἔνθεος ’; compare Old English ylfig insane, lit. ‘elf-possessed’, similarly < ælf elf n.1
Early alternate spellings include "gidie," "gidy," and "gidi."
By 1556, "gid" was a term for a brain illness in sheep, likely related to the implication of insanity. A source cited at 1745 offers "giddiness" or "gid" as a term for this disease:
- Agriculture Improv'd: Or, The Practice of Husbandry Display'd by William Ellis
The other Day you lost a Sheep by the Gid, or Giddiness.
Nothing I could find in OED appeared to indicate that "Gid" was a precursor of "God," though it seems to have been an alternate spelling at some points in time.
God:
Cognate with Old Frisian god, Old Dutch god (Middle Dutch, Dutch god), Old Saxon god (Middle Low German got, (inflected) gōd-, godd-)
According to Bosworth-Toller Anglo-Saxon Dictionary, "gid" or "gyd" meant a poem, song, proverb, or riddle earlier than the meaning related to sheep illness.
Gid oft wrecen (a song oft sung) [recited], Beo. Th. 2135; B, 1065.