This claim is true, along with some clarifications. Not all republicans are banning drag, but drag performances are now under scrutiny and risk of bans in specific states across America.
Just to make sure that your information is as clear as possible, drag performances/shows and drag races are different things. Notably, a drag race is a type of motor racing in which motorized vehicles race to a finish line. (Britannica) This is not to be confused with Rupaul Charles' award winning television show "Rupaul's Drag Race", which helped to popularize the art form of drag in popular media and among straight audiences. For the rest of this answer, drag in terms of this issue are referred to as drag performances or drag shows, as the show itself is not being banned by lawmakers and politicians.
In March, Time reported that "Tennessee became the first state to explicitly ban drag shows in public spaces on Thursday after Gov. Bill Lee signed the provision into law hours after the measure passed in the state Senate." (Time) This is the law that is garnering the most attention at the moment, but is not the first seemingly anti-LGBTQIA+ legislation to pop up recently. Time went on to describe how bills alike to this one have been proposed in at least 14 other states. Not all of these bills have not passed officially like Tennessee's has, but these bills have all come from conservative and right-wing politicians. The common theme of these bills is a message of "protecting children" from drag performances. In their phrasings, drag performances are compared to cabaret, specifying that they are any times in which a person is impersonating the opposite gender.
As of March 5th, 2023, Arkansas and Tennessee are the only states who have officially signed their anti-drag show legislation into law, while Arizona, Idaho, Kansas, Kentucky, Missouri, Montana, Nebraska, Oklahoma, South Carolina, Texas, and West Virginia all have similar bills in varying degrees of approval in their respective courts. In the coming months, it is presumed that these bills caught in limbo will fall through and pass into state law as well.
Britannica source: https://www.britannica.com/sports/drag-racing
Time source: https://time.com/6260421/tennessee-limiting-drag-shows-status-of-anti-drag-bills-u-s/