Even with the perfect teacher for him, your son would be undergoing unusual stress this school year; add in a burned-out eacher like this one to the mix, and your son's stress and anxiety levels are probably in the stratosphere.
Since you say you don't know what to say or do at this point, please, use the resources your school has for ALL parents! I'd involve the school psychologist and/or counselor immediately. You need some objective, third-party eyes on this ASAP for a couple of reasons.
First, you need someone to observe your son -- not just a few minutes on one day but several times -- and you should push for that professional to meet with your son too. If he and a counselor or psychologist "click," he might benefit a lot from "going to see Mrs. Jones" a few times a week. Second, you should meet with the counselor or psychologist without your son there, to explain the home situation and the poor fit of your child with this teacher's style (and be frank -- yes, you and this teacher are work colleagues and the counselor/psych works for the same school, but your talk will be confidential, and you need to be really blunt about how the teacher's specific style is stressing out your already stressed son). Third, the counselor or psychologist needs to be willing to work on your son's behalf with the teacher--you may need to request a meeting with teacher, counselor and you all there, and that could get touchy; the teacher isn't very likely to allow more moving around in class just because one kid has issues (that's how she might see it) because she'd have to change her style for the entire class. But the counselor should be there to help navigate this with you.
I note that your post only mentions how YOU can deal with your son but does not say whether you have already talked with his teacher about your son's problems, or gone to the counselor. I'm wondering if maybe you don't mention the teacher or counselor because you're concerned that--especially as you are new to this job--the teacher will label you, and by extension your son, a troublemaker--? Any chance that you're worried about how it will be perceived if you as a parent have to criticize someone who is also a work colleague? I think the counselor could help you with that because surely the counselor has seen that before.
BUT I also wonder if even that is enough. You seem to know your son well, which is terrific, and you know that he is probably a "kinetic learner" or at least a kid who needs breaks to release his energy during the day. Knowing that, and knowing this teacher's style much better than a non-teacher parent would....Is there another second grade teacher at the school who would be a better fit for your son? Is there one who will recognize and work with his need for movement? One who can handle young boys' distractedness and, sometimes, lack of impulse control without always treating those things as "bad behaviors" to be punished?
I'd move him ASAP to another second grade class if there is another class and you and the counselor feel it would work better for him. But if there is no other class or you get barriers put up ("We just can't do that...") can you move schools? That is easier said than done, especially if you are the transportation for all your kids to and from school; also, you may not be allowed to transfer a child on the basis of "We feel my child should not be at the school where I'm teaching" (which isn't the real situation here, but they are not going to let any parent transfer based on "My child's teacher has a teaching style that's stressing out my kid"). But find out if you CAN do it.
Before you think about a transfer, though, do go the psychologist/counselor route and get your son some help to deal with his loss--because he's probably seeing himself as having lost you to your two new jobs at once. He's grieving the loss of that time and the loss of what was his usual, normal way of life that he'd had since he was born.
You also want to take care that your son doesn't end up seeing a change of schools this way: "I'm so dumb that mom is throwing me out of HER school so I must be really bad or stupid and she wants to get rid of me...." You can tell him over and over that those things are not true, but what matters is how HE perceives things, and that kind of negative perception sounds like a real possibility, considering how he's thinking right now. Again, there are so many reasons to get the professionals involved, and there should be professionals right there at the school or at least available through the school system.