With regard to the "a" as in "cry-a" in this song, that letter is pronounced in this song-and usually pronounced in African American songs and rhymes as "ah".

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I think I'm also hearing:

6.Oh, I b'lieve to my soul she had a lucky HAND
'Cause the lady gave HER thing to the street car man.

But it could be "sweet talk man" (a man who talks sweet, "gives a lot of compliments, flirts, says "sweet nothings")

"Lucky hand" could mean cards, or it could mean being generally lucky in life, or it could mean something else.

As to her "thing", well you know what that means.