Here's the in depth fridging guide:
https://www.caudata.org/forum/showthread.php?t=85263
Fridging is usually the fix for floating, so hopefully it helps! Also it'll keep your axie safe while you get the nitrates under control.
Have 2 tubs for fridging. He will be in one for a day, then the next day set up the new tub (keep dechlorinated water in the same fridge so it's the same temp) and pop him in. Then you can clean the first tub ready for the next day.
A super fine net is probably the best way to transport axies, but I personally find it easier using my hands (sanitised). It does depend on the axolotl as well. My wildtype is fine with me using my hands but my copper doesn't like it so I use a net with him. As long as your hands are super clean (any parts going in the water needs to be sanitised), you're gentle and using both hands it's okay. You want to especially watch for their gills. If you're not comfortable using your hands then a net is great!