How do I get my dog to stop whining in public?

You can start by making sure you get his attention with treats, when there are kids or exciting stuff happening outside. I believe that it's a good idea to reinforce him into a calm state where he doesn't prance around or whines, where you get his attention inside the house and reward him with a treat once he sits down. You can then raise the training level by making him not just sit, but also lay down, and make sure that he calms down a bit before you reward him for his calm state. With all kind of training with dogs it just requires patience, practice, consistency... do this for a few days, maybe dedicate 10 minutes to each session where he starts to whine, don't get frustrated if the behavior doesn't go away on the fly, it might take a couple of sessions, as long as you get him to calm down once or twice per session the number of times you need to calm him down will reduce over time for sure. You will notice that by the time you take your training to a public area such as the store how you describe you will have a much calmer dog that will probably won't need a lot of effort because you will notice that the dog will most likely be calm because of the behaviors he learned while being in the house. And just in case for public areas: You can start to determine if the bad behavior is cause by "you" the master leaving his sight by having someone stay outside with him for a bit to see how he reacts to you leaving.

We built up a settle in the home environment on a specific mat that he has learnt to settle on at home where there are no distractions. We made an effort to visit pubs and cafes when they were QUIET and we usually opted to sit outside with him if at all possible. We made sure he had a huge walk before-hand so he was at least a little more tired and ready to settle. We would sit down and ignore Byron, until he would eventually lie-down. As soon as he was down, we would very calmly and slowly put some treats between his paws. We then started to reward when he wasn’t looking at us and when his head was naturally resting between his paws. We then left longer and longer between each reward. One thing to note was the definite extinction burst in his whining and pawing behaviour. It absolutely got worse before it got better and he cried louder and pawed harder at us, but because we stayed strong and consistent it is much less prevalent and the lie-down with head-down position is beginning to be his go-to behaviour.