Have you seen the help pages?
Download, install, and authorize—ArcGIS Pro | Documentation
Frequently asked questions—ArcGIS Pro | Documentation
It wasn't clear from your post
I have exactly the same problem and ill be honest the help on what you actually need to do to fix this issue is very poor and unclear. The instructions provided in the onboarding information don't match the actual webpage you are faced with when you click on links. I have read through the instructions linked above and its not actual helpful on how you change your account from public to an organization.
Completely agree with the original posters experience.
Can anyone provide any guidance on how to actually change my public account to an organisation so that I can log in to the single license that I purchased. Super grateful if anyone can help untangle this one.
if anyone gets stuck here - found the solution - it really comes down to ensuring that when you go into the initial link with the instructions - bottom right there is an activate subscription - this then lets you change your account from public to the organisation you can set up. works like a charm now.