AUSTIN, Texas—Returning to Texas after a lackluster campaign for president, Gov. Rick Perry faces a diminished role on the national stage as well as trouble at home from unhappy voters and Republican upstarts eyeing his once unassailable hold on power.

The longest-serving governor in the history of the Lone Star State remains its most potent public official, with a large campaign war chest, solid backing from social conservatives and scores of political appointees who remain loyal to him.

But...