HOUSTON — First, Judge Sandra Watts was stopped while trying to vote because the name on her photo ID, the same one she had used for voter registration and identification for 52 years, did not exactly match her name on the official voter rolls.

A few days later, state Senator Wendy Davis, a Democrat who became a national celebrity after her filibuster over a new abortion law, had the same problem in early voting. So did her likely Republican opponent in next year’s governor’s race, Attorney General Greg Abbott.

They were all able to vote after signing affidavits attesting that they were who they claimed to be. But not Jim Wright, a former speaker of the House in Washington, whose expired driver’s license meant he could not vote until he went home and dug a certified copy of his birth certificate out of a box.

On Tuesday, Texas unveiled its tough new voter ID law, the only state to do so this year, and the rollout was sometimes rocky. But interviews with opponents and supporters of the new law, which required voters for the first time to produce a state-approved form of photo identification to vote, suggest that in many parts of the state, the law’s first day went better than critics had expected.