Answer
No, Gohmert did not win the 2022 Texas Attorney General election. The winner was Ken Paxton, who was re-elected to a third term in office. Gohmert was one of five Republicans who ran against Paxton in the primary election, but he finished in fourth place with just over 10% of the vote. Paxton went on to defeat Democrat Rochelle Garza in the general election by a wide margin.