In "dodgery" one can see the verb dodge, meaning "to evade by a sudden or repeated shift of position." That verb entered English in the 17th century, but the noun form of "dodge," referring to the act of evading something by such a motion, is over a hundred years older. "Dodgery" entered English around the same time. You can dodge things physically - as in getting out of the way of the ball in the popular schoolyard game - or figuratively, as in shirking one's responsibilities, often by some kind of deceitful action. From there the use of "dodgery" to mean "evasion" or "trickery" is hardly an elusive connection.