By definition children have little "self control". If they did they would be little adults!!! It will not please you to hear it, but you have a "perfectly normal" class in many respects. You have several issues happening simultainously.
1) They have been sitting ALL day long, this is likely (LITERALLY) the first physical anything they have done or had since their heads left the pillow that morning. Suggestion one, DRAIN them physically for the first 10 minutes, hopping, running, whatever you can imagine. If it has a martial component/sneaky context all the better. Have them do 100 punches.
The idea is to give them a physical outlet, get them into their bodies, and out of their heads. Having them do the big gross physical motions is part of what you are trying to do right?
2) The age range is too BIG. You cannot put kinder garden kids with second, third graders and try and work with them as a single group. It will not work for the kids on either end of the age range. The "games" used for the kinder gardeners are not the right ones for the 2nd, 3rd graders.
You cannot teach the same way to that big an age difference. Very different approaches... same information, just a different speed.
You have to split the group, or divide the exercises so they you have lots of circle drills. (you or one of THEM in the center). If you're losing just one or two when you have to waste time attending to them the entire group is not being taught. In a line that can be bad. In a circle kids tend to do less because "everyone" is able to see them doing "X".
3) As for the talking, there are many approaches, if you can make YOURS work, you win. Ever tried the "NOISE game"? "...Ok class you can make as much noise as YOU want UNTIL I raise my hand.... after that you must be absolutely 100% quiet or you sit down. The winners are the ones standing. Run 2,3, 4 rounds of it.... Should get you 5 (maybe 10) minutes of less chatter. Perhaps more?
4) Have you spoken with any of the teachers of the school re: their specific protocols for disruptive behavior(s)? Several places I teach have them and I borrow theirs. We are a bone fide class, genuine teachers not ~...just the hour they can fool around... and learn pretend karottty~
5)Also remember we cannot have ADULT expectations of the kids. They chatter. You should hear what is happening with the COLLEGE students today. They literally can no longer be silent or still either.
6) As for the touching that's an easy fix. Most gym teachers have little squares/circles/triangles. You can use paper plates if need be. Or you can cut up your own pieces of a sticky mat, etc. You put them down, such that the kids physically CANNOT touch one another sitting/standing on them. Place them far enough apart.
Touching (without permission/instruction) gets them in time out IMMEDIATELY. 1 minute per year age of the child. If they get in time out you send home a standard form note, and ask the parents to sign it. ".... little johnny was being disruptive in my class today we thought you would like to know about this. Would you please remind johnny about correct classroom behavior..."
Or you can write a fast note specific if the behavior was disruptive enough. "...Maryjane refused to stop talking for 8 minutes straight (and continued to defy me by talking even while sitting out) and disrupted my class... please speak to her about appropriate behavior for my class"
7) The other likelyhood is that you could well have multiple children with assorted medical/psychological diagnosis. ADD, ADHD, etc., etc. Regardless of the particular label, the behaviors follow a certain generic pattern.
Are certain children the consistant problem for you? Are there specific behaviors, beyond the chatty and touching you described?
Where are things falling apart in your intended curricilium? If you can give us an idea what it looks like perhaps others can help as well?
Lastly, are you familiar with "spring fever"?
When time allows,