You're in a similar situation to me right now, I have a 30 gal and a 10 gal that were set up within the past month. My 10 gal was full of particles last night, it looked like a snowstorm. I took the axie out and changed almost all of the water, and it looks a lot better today. By the way, these particles shouldn't be harmful to your axie, the ammonia and nitrites are the main problem. Here are a few things that have worked for me so far:
Your levels are likely caused by the fact that your tanks are still cycling (growing the bacteria that naturally remove toxins and waste). Because of this your levels will be unstable until the cycle is complete. It can take 4 weeks or more for the good bacteria to establish themselves and get the toxins under control.
There are a couple of things you can do to get rid of or reduce the particles floating in the water. First of all, next time you do a water change, take the filter inserts out and rinse them in TANK WATER (not tap water!) The filter is the main place where the good bacteria grow, tap water will kill those bacteria. The same applies if you decide to rinse other items in the tank, the bacteria grow on all surfaces. Don't clean the glass on the back and sides of the tank for now, this adds another surface for the bacteria.
Secondly, stir up your substrate thoroughly and let it settle before you vacuum the tank. This will get rid of more waste, including waste that can collect in substrates and foul the water. It's a good idea to stir up sandy substrates regularly.
Third, do a water change of at least 50%, up to 90% if the levels are very high or there's a lot of waste in the water. This will clear out more particles and lower the levels of toxins. You don't have to do this every time, just once for now, If you do a 50% water change this only removes 50% of the toxins, so if they're really high it's good to change more of the water.
I do partial water changes of up to 50% daily because my levels are still high and so is my pH. I just bought a water changer, it's my new best friend.
It generally takes weeks for a tank to cycle, but check your toxin levels every day to decide if a water change is needed and how much water should be changed.
If your levels are high and you're concerned about your axie, you can also add a tablespoon of aquarium salt per 10 gallons, once a week. This helps to buffer the effects of ammonia and nitrites on your axie.
Personally I'm not a fan of adding too many chemicals to the water, but some people like to use stuff like API Stress Zyme or Ammo Carb, bacteria supplements or pH adjusters. My experience with these is that they have more effect on my wallet than my fish tank! Usually the bacteria supplements are the wrong species of bacteria. You can also get filter inserts that absorb ammonia. I'm using one but not noticing a lot of difference so far.
You can promote bacteria growth and clean the water with live plants. I have Marimo moss balls in my tanks and I'm planning to add more plants. You can increase the oxygen by adding a small air stone, you can get little $2 valves to regulate the air flow so your axolotl won't get stressed by water flow. in my 30 gal tank I have a "hang on back" power filter and a sponge filter driven by an air pump. The sponge filter is mainly for growing bacteria and it filters particles from the water.
I think your filter hangs on the back of the tank as well? If it's creating a lot of water flow you can get a suction cup soap dish at a dollar store and stick it to the glass right under the water outlet. I even put a little filter insert bag for ammonia on the soap dish so the water flows through it. It works great and has never fallen off.
