Jones also opposed gambling and urged a revised criminal rehabilitation program for Louisiana.
But chiefs who oppose gambling said the crackdown will help the reservation's residents.
Mr. Cornyn, however, already had opposed gambling, according to a spokesman.
While some of those who oppose gambling have talked of returning to a traditional economy, most take jobs off the reservation, often in construction.
Mr. Fletcher has said he opposes gambling, but would be willing to hold a referendum on the issue.
He has consistently opposed casino-type gambling in the state.
Mrs. Mitchell's husband, who opposes gambling, left the reservation with their baby, she said.
Some Seneca factions oppose gambling, while others say the leaders are not driving a hard enough bargain with the state.
The church also opposes pornography and gambling, including government- or charity-sponsored lotteries.
Roberts also opposed expanded gambling in the city and the 2010 Olympics.