The case of a Texas woman who sought a court-approved abortion but wound up leaving the state for the procedure is reigniting political arguments that have roiled elections for more than two years, placing Democrats on the offensive and illustrating Republicans’ continued lack of a unified policy response or clear strategy on how to talk about the issue.

The Texas woman, Kate Cox, a Dallas-area mother of two, has emerged as the living embodiment of what Democrats say remains one of their strongest arguments heading into the 2024 election: that Republicans will ban all abortion. Ms. Cox was more than 20 weeks pregnant with a fetus that had a fatal genetic abnormality known as trisomy 18, and lawyers and doctors argued that carrying the pregnancy to term put her health and her future fertility at risk.

Her lawsuit was one of the first attempts by an individual woman to challenge the enforcement of abortion bans put in place by Republican states after Roe v. Wade was overturned a year and a half ago. Hours before the Texas Supreme Court ruled against granting Ms. Cox a medical exemption to the state’s abortion bans, she had decided to travel to receive the procedure in a state where it remained legal.

From top officials on President Biden’s campaign to candidates in battleground states, Democrats jumped on Ms. Cox’s plight as a cautionary tale for voters next year, highlighting her situation as they have done with the wrenching, deeply personal stories of other women and girls since Roe was overturned.