had had its share of 是用 过去完成时表达 "过去确曾也有过" 的意思.
have one's share of sth 说白点就是 "没少出现过xxx" 之意
马萨诸塞州之前确曾也有过不诚实（或虚伪不实）的政客
或者
马萨诸塞州之前没少出现过不诚实（或虚伪不实）的政客