If you get the testing and the scores are high enough, you can connect with a Davidson mentor.
When you call the psychs, talk with them a few minutes and see who seems helpful, who is seeming to really get your child. They will vary. The right fit for one person could be a poor fit for the next.
The book Bright Not Broken I suggested is specifically on this overlap in behaviors between gifted, ADHD, ASD. It sounds like he really responds well to instruction, so some books, just a little bit to bump his connecting the dots on social thinking would probably go a long way. For instance, maybe not this particular book, but a book like this could be just enough to help him realize what's going on Interrupting Chicken: David Ezra Stein: 9780763689032: Amazon.com: Books Or there are full curricula like We Thinkers.
Your library might have access to books like this, at least on inter-library loan. And if you click that one on amazon, there will be more in the list of books customers also looked at. You'd probably find something helpful.
That's really cool that he's doing so well in karate. Coaches enjoy working with my kids too. They like how capable the kids are of understanding really precise verbal instructions. Given that he's bored, even with that and school, you want to consider stepping up his access. Not in a pushing way so much as satisfying. Like my dd, bright, not necessarily a gifted IQ, and with homeschooling her at that age she was doing latin, learning history, all sorts of things. School just really holds these kids back. It actually physically HURT her not to have access to the things she wanted to learn and do.
With my ds, his intellectual side, his brain, just just has this ENERGY. It's a drive to develop, and that energy has to go SOMEWHERE. So right now in your ds the energy is coming out with talking, but it could be going into learning spanish. It could be going into doing science kits or working through TOPS science books. He could be doing pre-algebra. If he is ready for multiplication, he's almost there. Get him the Hands-On Equations app and see what happens. I'm just observing that the level you have him in is NOT using up his developmental energy, the energy that is telling his brain DEVELOP, DEVELOP. It needs healthy outlets.
I'm going to read Bright Not Broken for certain! That looks like something that I definitely need to pay some attention to. This weekend we've decided to lift the math ban and just let him read about whatever he wants. I gave him a 1st grade workbook today for the first time after school and let him work a couple pages and he did fine, not one error...my fear prior to this has been making him confused with different work but after all this we are just done trying to worry about it. I'm going to do some digging on homeschooling and it will replace homework time. We are still going to have him tested though, when he turns 6....we already talked to a couple of options and the one we like (so far) is from the site you recommended above...Hoagies Gifted. By the time we get him in, he will probably be about finished with Kindergarten so we will have a new teacher, and he should be able to start 1st in a better place. I downloaded the Hands-On Equations app just now also, and we will let him try his hand at it tomorrow....I found several good ones that were related, so I figure what the heck lets see what he can do.
These past couple days have been exhausting. I cannot thank you all enough for listening and giving advice without judgement.