SILSBEE, Tex. — The last time a Democrat won Texas in a presidential election was 1976, when voters chose Jimmy Carter over Gerald R. Ford. Carey Georgas was 22 back then, a disillusioned Democrat who voted for Eugene McCarthy, the independent candidate, and who was working at Cravens Insurance in this East Texas town.

Forty years later, Mr. Georgas, 62, is still a Democrat, but not so disillusioned. He says he is voting for Hillary Clinton and has been following the polls showing Donald J. Trump with a narrow lead over her in Texas, one so slim that it fell within the margin of error in some surveys.

Yet, like many Democrats in a state that has voted to send Republicans to the White House in nine consecutive presidential races spanning four decades, he is equal parts optimist, realist and pessimist. Many of his friends and neighbors are Republicans who are supporting Mr. Trump.

“I’m bumfuzzled by the whole thing,” said Mr. Georgas, now the president of Cravens. “I don’t think she’ll win the state. But I think she’ll close the margin closer than anybody has.”