If you can afford it, can you take him to a chiropractor or naturopath? Most chiros/NMDs that I have worked with charge around $99 for the first session and then around $30-40 for each follow up visit. When my oldest (now 16) was first diagnosed with ADHD I was not ready to put him on medication because he was young (7) so we worked with a naturopath for a while. She came up with a plan of supplements and dietary changes for him. His supplements were fish oil (which everyone can benefit from), a supplement for adrenal support (Drenamin,you can only get it from a clinic licensed to sell Standard Process products) and a few others. He was noticeably calmer and more pleasant, with better sleep and fewer outbursts, immediately. We ended up not sticking to the plan because the office was an hour away and I just couldn't consistently carve out 2.5 hours a week in the middle of the week for visits, and the supplements were expensive. They definitely helped though and something like this might be a good option in the short-term for your son. Another thing that my son now takes (and has taken for a while) is Juice Plus. He calls them his happy pills and it's the one thing he makes sure that he starts the day with. He says that he feels "off" and agitated without it. You have to buy that through someone who sells it - you can't get it in stores - and it's about $45 a month and they automatically ship it to you and bill you each month. I was skeptical about it but he really likes it so we keep using it.
Magnesium is also very calming, B vitamins are good for mental focus and are also calming. Almost everyone is deficient in Vitamin D. Dietary changes can help a lot too...a breakfast high in protein and fats (eggs, sausage, a smoothie made with protein powder, nuts) will help avoid a mid-morning carb crash. For some kids, eliminating inflammatory proteins such as gluten and dairy products, reducing sugar consumption and artificial colors and flavors makes a big difference in how they feel and act.
The thing is that when you start venturing into nutrition and supplements, changes can get expensive quickly and you might be spending your time and energy on a strategy that doesn't work for your kid. That's where working with a naturopath or chiropractor can end up saving you in the long run. You'll have someone who can use their diagnostic expertise to say "in other kids with similar issues, we've seen that these three changes have been the most effective: X, Y, Z" and then you'll know to try those first instead of 5 other things that might not be right for him. If finances are an issue, be upfront with the practitioner and let them know that you don't have an unlimited budget in terms of time or money, that you're not grocery shopping at Whole Foods and need to really target your changes to those that will have the biggest impact. They should still be able to come up with some suggestions that can help your son.
Good luck to you!
ETA: others have suggested an evaluation at school. While this is necessary to assess baseline metrics such as cognitive ability, behavioral issues, etc. it's of course only one piece of the puzzle. A school evaluation CANNOT diagnose or treat medical problems such as ADHD, anxiety or mood disorders. You are right to pursue having him evaluated for those issues through his doctor. There are components that overlap and having the school's testing and observation results can help with the medical diagnosis, but it's not a substitute for that medical diagnosis.