How do you deal with talking on your feet or answering tough questions, especially during presentations?

How do you deal with talking on your feet or answering tough questions, especially during presentations? I work often have to give presentations for work, occasionally on sensitive, political, or controversial topics, and I find that on top of dealing with the challenge of doing the presentation I have to answer tough questions. This has posed a challenge because it's hard for me to think on my feet and essentially "bullshit" my way through answers, which I feel is part of my job. So, for example, say we're cutting funding for a popular program I explain the why, the logic, and the data, but have to answer tough questions from the audience "why can't you cut this other program? why are we losing money, etc". I know I need to deliver the answer in a customer friendly way without taking a side, but I often freeze and stutter my way through it. To be honest, I think people feel bad for me and don't ask me to elaborate cause they think it's rude or don't want to go through the ordeal again. In contrast, my colleagues can face the music and bullshit/lie their way through these kind of presentations and I feel very inadequate. How can I get better at presenting and thinking on my feet? On top of dealing with the stress of the presentation, I have anxiety on trying to answer tough questions.