Acrobat X document saving problems on Mac OS Sierra.

One more thing regarding the new UI: Nobody I know who came from an earlier version of Acrobat likes the new UI (including me, if I had a choice, I would go back to the Acrobat 5 UI ). What usually helps in getting around the UI is to use the tool search function: You don't have to know where a tool is located, as long as you remember it's name. Then just type in the tool name into the search field at the top of the right hand panel, or at the top of the "Tools" area, and that will give you direct access to that tool, or the group of tools it is located in.

Yes, Acrobat X is no longer supported, but what's more important is that Acrobat X was never supported on Sierra - it's supported ended long before Sierra was even announced. It's one thing to try to get Acrobat X to run on an operating system that it was supported on by the time support ended, but something completely different to assume that Adobe had a time machine and was able to test the software on a future version of the OS

You very likely won't like this answer, but Acrobat X will never run without problems on Sierra (just as Acrobat 9 is not running correctly on e.g. Yosemite and El Capitan). Adobe will not go back and fix what's wrong in Acrobat X to run on Sierra, and Apple will not change Sierra to make Acrobat X like it it on older versions of the OS. ]

You have a few choices: Downgrade the operating system so that Acrobat X runs on something that it was meant to run on, upgrade your Acrobat version (unfortunately, you are no longer eligible for a discount when upgrading), or install a virtual machine and run Acrobat X in that VM on an OS that is supported. I use the latter approach for older versions of Acrobat, running in Parallels Desktop.

are the files write-only on that server?

if not, you could try to reset your preferences -   https://forums.adobe.com/thread/1595848?start=0&tstart=0