Is "sore" the origin for "so"?

Solution 1:

According to Google Search [ define sore ],

sore adverb archaic 1. extremely; severely. "they were sore afraid"

This use of the word sore is obsolete in modern English but still familiar due to its frequent use in in the 1601 Authorized Version of the Bible, such as the passage you cite from the Christmas story, which is well known to anyone who watched the Charlie Brown Christmas:

⁸ And there were in the same country shepherds abiding in the field,
keeping watch over their flock by night.
⁹ And, lo, the angel of the Lord came upon them,
and the glory of the Lord shone round about them:
and they were sore afraid.
(Luke 2, AV)

In modern English, we still use sorely in nearly the same way, as in “he will be sorely missed”. And in modern German, sore has a close cognate: sehr, which is often translated very but in the right context can mean sorely. For example, “Ich werde ihn sehr vermissen” means literally “I will sorely miss him”.

Online Etymology Dictionary derives sore from

Old English sar "painful, grievous, aching"

but so from

Old English swa, swæ (adv., conj., pron.) "in this way"

and the two words do not have common ancestry.

Solution 2:

To answer the question very simply: no. ‘Sore’ is not the source of the intensifier ‘so’.

‘Sore’ was already explained in Ed’s answer; and ‘so’ is a very old word. It is one of the many words that derive from the third person pronoun stem *so-/to- in Proto-Indo-European (i.e., about 5,000 years ago). In Old English, it was swa, and it is cognate with similar words in all the other Germanic languages. It is, in other words, a much older word than ‘sore’ in this intensifying usage.