In the only statewide race on the ballot, ex-legislator and evangelical speaker Rick Green faces longtime family law judge Debra Lehrmann in a Republican runoff for a Texas Supreme Court nomination.
The winner faces Democrat Jim Sharp in November.
In other races, voters will pick between two GOP State Board of Education candidates. They'll also decide party nominees in seven congressional runoffs and nine legislative runoffs.
Many Democratic and Republican primary contests, including nominations for governor, were settled March 2.