“It shaped who I am today,” she said. “A lot of people ask, ‘Is she tough enough?’ West Virginians know me as someone who can stand up for what is right.”

After leaving her office in the State Capitol last week to pick up her 11-year-old daughter from school, Ms. Tennant sat for a quiet-voices interview in the children’s section of the main branch of Charleston’s public library, while her daughter studied for a test.

She framed her candidacy as a contrast with “the dysfunction that Congresswoman Capito has continued to perpetuate” in Washington, citing the government shutdown in October.

Ms. Capito, 60, the daughter of a former governor here, said in an interview that she had opposed the shutdown and that she voted for the recent budget compromise that Tea Party-leaning Republicans rejected.

“I’ve been known as someone who reaches across the aisle,” she said. She spoke by phone from her home in Charleston, where she was readying an office Christmas party for her husband, Charles, an investment manager of Wells Fargo bank.

Ms. Capito is well liked by leaders of both parties in a small state where most people in public life know one another. Polls show her winning up to one in three Democratic votes. “I couldn’t be elected without Democrats,” she said.

More than in any other state, deep ties to King Coal expose Democrats’ vulnerability in West Virginia because of the national party’s stances on climate change and renewable energy. Both Ms. Tennant and Ms. Capito diverge sharply from the Obama administration on its coal policy, an issue that is often more emotional than economic in Appalachia, where jobs are being lost to cheaper natural gas as much as to environmental regulation.