The reason you think you hear "to" with a short "O" (as in "not") is because the speaker is actually completely omitting the vowel all together. He's saying, "I am going t'town" - and you think you are hearing "ta town."
If he were to say, "I'm going to the store" you will hear the proper "to" - simply because it's awkward for the speaker to say anything like "I'm going t'the store." If he were to really run the sentence together, it would be more like "I'm goine t'thtore" - but that is almost like a lisp and you wouldn't pick a short "o" out of that.
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