DALLAS—American Airlines is banning emotional-support animals in a move that will force most owners to pay extra if they want their pets to travel with them. The airline said Tuesday that it will allow animals in the cabin free of charge only if they are trained service dogs. The change takes effect Monday, although passengers who already bought tickets can fly with a companion animal until Feb. 1. The move follows a similar decision by Alaska Airlines and is likely to be copied by other carriers. The U.S. Transportation Department cleared the way for the crackdown against companion animals last month. American said passengers with a service dog will need to complete a government form vouching for the dog’s health, training, and temperament. Other animals, including dogs not trained as service dogs, will only be able to fly in the cargo hold or a kennel that fits under a seat …