Anne Wilburn, 61, a native Texan who lives near where Davis spoke in Houston, said she could relate to her story and planned to vote for her.

"I certainly don't think an accent and a cowboy hat and graduating from Texas A&M makes you more Texan," said Wilburn, who works part time from home as an office administrator.

Davis' effort has required distancing herself from more liberal supporters — many from out of state — who flocked to her after the abortion rights filibuster. During her appearance in Houston last week, she expressed support for allowing people licensed to carry concealed handguns to carry them openly as well, though as a member of the Fort Worth City Council she supported gun control measures.

She also said she would not oppose a ban on abortion after 20 weeks of pregnancy, and backtracked on past support for legalizing marijuana.

It's not clear whether these policy shifts will motivate or repel voters. Jeremy Bird, a founder of the Democratic advocacy group Battleground Texas, said its targets were Latinos alienated by Republicans' stance on immigration and voter registration as well as suburban women troubled by the party's position on abortion and education funding — its recipe in other conservative states that are becoming more moderate.

Wendy Faith, 39, a teacher and mother of three in the Houston suburb of Katy, is an independent voter who favors abortion rights and some gun restrictions.

Faith likes Abbott, but is upset by Texas Republicans' support for voucher programs and cuts to school funding. She also likes some of Davis' more liberal approaches, but doesn't know her as well. To this native Houstonian, both candidates seem like slick politicians.

"They have to become approachable," Faith said as she waited in a school carpool line last week.

Neither candidate has her vote yet.

molly.hennessy-fiske@latimes.com