Fuel Delivery problems of the 2005 Chevrolet Suburban

Last week, I filled up my gas tank as usual. When I went to pull off, the gas light failed to go off, the gas needle did not move and the bell to indicate that the tank was empty went off. The dealer stated that it's my fuel sending unit and that he's seen several of these defect lately. . . And that I should check for a recall. Since, I've had to monitor when I gas up and how far I drive. . . Which is not only frustrating, it's scary! the needle now floats from full to empty and everywhere in between, but for the most part, it rests on e. Vehicle in question is 2005 Chevrolet trailblazer. . . This model is not given as an option in later screen where I entered "suburban". I own a chevy trailblazer. Thank you.