I still cannot sign into my Apple ID or iCloud on my Mac. Still says server error

Are you running any anti - virus programs? Using a VPN?

Try setting up another admin user account in System Preferences/Users & Groups to see if the same problem continues. Please post back on whether or not this worked. Also try the Safe Mode. Please post back on whether or not this worked.

Isolating an issue by using another user account    

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If it works in the Safe Mode, try running this program when booted normally and then copy and paste the output in a reply. The program was created by Etresoft, a frequent contributor.  Please use copy and paste as screen shots can be hard to read. Click “Share Report” button in the toolbar, select “Copy to Clipboard” and then paste into a reply. This will show what is running on your computer. No personal information is shown. If the log won’t post, try posting it in Pastebin and provide a link in a reply. After pasting the report in a PasteBin page, go to the top of the page, and copy the address in the URL bar. Paste that in a new reply.        Pastebin.    

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There are so many problems at least similar to your that I have ceased trying to sort them out. I suggest you do this search (server or connection error apple) or use my search link server or connection error apple. You will find many read that may be connected with your problem. I emphasize that you may want to use the support options below in dealing with your specific problem.

Get a call from Apple support to help solve your problem using this

link https://getsupport.apple.com/?caller=kbase&PGF=PGF63005&category_id=SC0245&symptom_id=23362

Apple ID Support https://support.apple.com/apple-id

Apple ID account security for country-specific and region-specific help https://support.apple.com/en-us/HT204169

Get Support: We’re here to help https://getsupport.apple.com/. Go through the options to find the best support option.

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Also note that Apple server maintenance may be what is causing your problem. A week or so ago I saw a post indicating that, per an Apple tech, Apple is conducting major [server] maintenance and that many services are down. The instruction given was to "try again tomorrow ... [and] maybe the [needed] server will be back online.

This explanation for your problem is running thin though.