My DS7 (this board goes by ages rather than birth order) in 1st grade sounds very similar to yours. It's wonderful that your teacher has picked up on the discrepancy between intellectual capacity and output. Our teacher picked up on it too, but probably wouldn't have said much to us if we hadn't voiced our concerns two months into the school year. Our DS does not have dysgraphia, and a dyslexia test over Christmas indicated he doesn't have dyslexia either, but his slow processing speed affects his reading fluency, writing speed, and ability to perform under time constraints. We have WPPSI testing showing the lower processing speed (50th-ish percentile) in relation to other subtests (98th/99th percentiles).
Our WPPSI tester recommended that we allow him to dictate to us throughout his school years, when teachers will allow, so that he can get out on paper the thoughts that are in his head without the writing/processing slowing him down. We're in the process of looking into a 504 for him, which the dyslexia tester recommended, and I should be able to report more on that after we meet with his 2nd grade math teacher about it tomorrow.
We decided to pursue a 504 for him because his 1st grade teacher already initially thought he was just being lazy when his writing output didn't match the abilities she sees, and I keep reading that dyslexic children are often labeled as "lazy" when they get to older grades and their smarts are no longer able to hide their learning issues. (Although DS7 isn't dyslexic, that seems to be the closest "labeled" LD to his processing speed issues and their effects on his reading and writing output.)
I typically spend a decent amount of time finessing my posts to make sure they say what I want them to say, but am pressed on time here. My main thought is that if your teacher is noticing these things this early, and if you have testing that corroborates her observations, pursuing additional evaluations that will narrow down what's going on will help you help your DS, and can result in getting accommodations in place now that may make a big difference once he gets to 3rd/4th/5th grade. When I shared the results of our dyslexia testing with DS7's 2nd grade math teacher, she said it suddenly helped her make sense of some of her observations, and told us that getting something written up into a formal plan would help the school place him appropriately from year to year so he gets adequate challenge and isn't labeled as "lazy." We wouldn't have been able to move forward with this without the dyslexia evaluation and its resulting recommendations.