How can I deal with my increasingly embarrassing mother in public?
Let's start by addressing your mother's health. Anytime you notice a striking shift in someone's behavior, particularly an elderly person's, you should rule out medical issues. It's easy to forget about medical issues, because they're often invisible. If your mother has not had a full medical exam recently, see if you can arrange for one. Early dementia, glandular problems, a change in medication and other conditions can lead to changes in behavior.
If medical issues are ruled out, try talking directly with your mother about this issue. Point out in an neutral, nonaccusatory way the changes you've observed in her behavior. If she agrees with you, and is open to help, that is where you can focus your energy. The two of you could, for example, develop some working agreements about how to handle the situation when you are out in public together. Perhaps you could signal her with coded language that she has crossed a boundary with strangers and needs to stop. It might be a good idea to agree on a back up plan in case she doesn't pick up the signal and continues her behavior. You could, for example, gently take her by the arm and lead her away. This approach would spare both of you embarrassment.
If your mother doesn't agree with you about her behavior and/or has no interest in changing, there may be little you can do, and I would remind you of the limits of your responsibility. I presume that the other people she speaks to are adults. You can let them take responsibility to end conversations that they don't want to participate in. As long it does not lead to an unsafe situation for your mother, you can choose to excuse yourself. If you walk away a few times and, in a private moment, tell your mother why, this may be the most effective way to get your message across.
I have this problem with my mother as well. She makes racist or sexual comments to people and is overly personal. Plus she is horrifically mean to my father who has Parkinson's, both in public and private. It's very depressing. I've tried to talk with her about it but she sees only as far as the end of her nose. Plus I live across the country and cannot afford to travel to see them except at their expense and pleasure (they can readily afford it). Last time I tried to discuss these things with my mother, she basically told me not to plan to visit again. My poor dad; it breaks my heart because he's a quiet, polite man, made even more gentle by the effects of the Parkinson's so her behavior is a complete non sequitur to his and, as I say, she's also abusive to him. Very sad.