This sounds a lot like when i first got my axies. One axie swimming wierd and upside down that died.
And a cycling problem after that. I got the advice to do only water tests and 30% daily water changes for a couple of weeks. Then less often.
And it actually all turned out great in the end. It took a lot of effort and time but fixed it, and nature took care of the cycling with time. My other two axies never got sick during.
I believe that trying to speed up the cycling might be hard if you're new to all this, I would never dare to try that.
Blinking gills doesn't have to be alarming. And red gills are normal as far as i know. They can turn bright red if your axie is active, and go white if they've been chillin out. So even if the gills would be red, it is normal.
And One of my axies has always had curled gills, and the other one gets it after feeding and goes back to normal after a while. So it's great stress signs sometimes but all this can be normal to.
My problem has always been to keep calm, me stressed has made me act in ways that stressed my axies more.
I'm still very new to all this. But i know how hard and stressful this must be for you and a plan to follow might calm you aswell.
I would say If your axolotl is acting normal, try not to worry and just give it time.

water changes, tests, check temperature, and keep it dark.
It's advice people here has given me many times and it's always worked.
Best of luck to you! And sorry for your loss.
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