Hi @Andy
On my server I could open sharepoint online site using sharepoint designer 2013 normally:
Could you provide some screenshot to show this issue or is there some error?
What's more, below is the detailed steps to clear cache for you to check (This is only applicable if the SharePoint Desinger is not a fresh install)
a) On the local computer, browse to the following folder:
C:\Users\<username>\AppData\Local\Microsoft\WebsiteCache
b) Delete all the files and folders that are present.
c) On the local computer, browse to the following folder:
C:\Users\<username>\AppData\Roaming\Microsoft\SharePoint Designer\ProxyAssemblyCache
d) Delete all the files and folders that are present.
e) On the local computer, browse to the following folder:
C:\Users\<username>\AppData\Roaming\Microsoft\Web Server Extensions\Cache
f) Delete all the files and folders that are present.
What's more, you could check your sharepoint designer 2013's version, here is a similar case for you:
https://social.technet.microsoft.com/Forums/office/en-US/24150607-ff54-4ab8-9ba9-d3175286de5f/sharepoint-online-best-practice-how-to-install-sharepoint-designer-for-connecting-to?forum=odspproductsandservicesharing
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