First, there is no need to be in a state of desperation. You have to maintain your equanimity if you are to be successful in dealing with behavior. Now, when you see your son starting to lose emotional control when with peers, immediately call him over and have him sit out in order to calm down. You'll need to do this over and over. It does sound like your son does get overwhelmed in groups of peers. He will improve over time. Try to avoid situations when he is with more than one or two peers for now.
There is not much to do re: kindergarten per se. It will run its course. All you can do is try to titrate his exposure to groups of children - the smaller the group the better. If you have any choice about the size of his grouping in kindergarten, select the smaller group(s). Who exactly is 'throwing out' the ADHD issue? Just as people should not casually suggest such a condition, you alos should not automatically reject it as a 'false diagnosis'. Should it be considered? Sure, if there is reason (e.g., excessive activity level) to do so. But it's reasonable to be conservative about use of medication if, indeed, after thorough diagnosis, it is concluded that he displays ADHD.
We are hoping it gets better over time, however, what do we do about Kindergarten? We cannot limit his interactions with other kids during this time and we fear that we will be getting calls daily about his behavior, which is sad because he is so bright. He is actually at a 1st grade level and we fear he will be so bored that he will act out. When he is engaged, he is fine. It is the transitions or waiting around that he has the most problems. People always throw out ADHD, but can it be something else? I do not want to be forced to get my son a false diagnosis so that he will receive extra help with his social skills. I really do not know where else to turn or what else to do other then hope and pray he matures before he is known as the kid who keeps bothering everyone.