Hi Geodaryl,
Yes that does sound like a fungal infection. The most cost effective, practical and least invasive strategy for you right now, is to fridge the axie with clean dechlorinated water for 3 weeks. You would need to perform daily water changes and monitor for progress or deterioration. If you notice any signs of deterioration despite a week of fridging, you would probably need to bring it to a vet.
I would actually recommend you commence salt baths at this stage. Besides fungal infections, salt baths are actually effective with ridding protozoa and ectoparasites and have some antibacterial properties. Salt baths are rather straightforward. Basically just prepare a tub of fresh dechlorinated water and dissolve 2 to 3 flat teaspoonfuls of non-iodized salt per litre of water. You can use salt such as
kosher salt, freshwater aquarium salt or sea salt. Do not use table salt. You can then place your axie in the salt solution for about 10 minutes but no longer than 15 minutes per session. Excessive salt solution soaking can cause damage to the axie's gills and skin. Have a look through this article as well.
http://www.caudata.org/cc/articles/salt.shtml
For the first 3 days, I would start out with 2 salt baths per day. Try to space them out as evenly as possible in terms of interval (ie. 9 am, 9pm) whatever best suits your schedule. An hour or 2 off is fine. There is no point for example in two salt baths in succession 3 or 4 hours apart. If you can gently tease off any flaking/peeling lesions that is also fine. You can do so gently with a cotton Q-tip. Roll the Q-tip towards you to gently lift away at the lesion rather than prod at it or roll inwards which can damage the skin.
Monitor the progress and response of the axie during the salt baths. From the 4th day onwards, reduce down to once per day until the end of the week. Continue to monitor. I anticipate a marked improvement by day 3. If by end of the week, the salt baths and fridging did not improve the condition, again a vet appointment will be necessary. The axie will struggle a bit in the salt bath. This is to be expected. Do not leave the axie longer than 15min. I would recommend 10min per soaking time.
You can prepare one or two bottles (1 or 1.5 L bottle) of salt solution in the fridge. The most convenient time to do salt baths will be during your daily water changes. As the salt solution is fridged, there will be minimal thermal stress.
Cheers.