Please see the Eligibilty option mentioned in the below link, which tells you what education proofs you need to provide:
You will not have to redownload the software.
Please refer to the below link to convert the CS6 trial into licensed software:
Let us know if this is helpful,
Welcome to Adobe Forum ,
You do not need to re-download, after purchasing the CC as per your eligibility & choice, you will need to log out & log back of the CC desktop App.
Next launch the product, it might give you a prompt to buy, in this prompt, there will an option to "license this software", click on that & sign in.
The CC will be activated.
In case you buy any other CS6 non subscription production, you will get a serial number, you would have to launch the download(if downloaded) or insert the DVD installer, you will get an option to run it as subscription or with serial number, you need to choose the second option, enter the serial number, sign in & it will be serialized.
Please let us know if it worked.