The official answer is that FM11 is not supported for Sierra. Not only is it not supported for Sierra but it's beyond end of life and not supported anymore at all. While you are not "forced" to upgrade, you will receive little to no help should something go wrong. And OSX is not really great for trying to downgrade back to previous OS versions.
There are a myriad of reasons why I would recommend upgrading. One is compatibility and support for modern OS, secondary would be more modern security features. FM11 is over six years old now, so much has changed in OSX since then.
Since I myself am not a "bleeding edge" early adopter of anything, let alone a beta tester, my standard practice is to wait a month or 2 after the new release of anything for all the other users to do the gamma testing, then I upgrade when the inevitable bug-fix X.X.1 release comes out. But I certainly don't let anything slide for half a decade.
Yep, Mike's right.
If you use FMP for anything production related and it's important to you or you're having system issues as you reported, upgrading is basically your only option for version 11.
Except that (if you don't need the new features or don't want to spend hundreds of dollars for a program that ... runs acceptably)... You could always try to find an old install program for a previous mac OS (where you didn't have issues with 11) and then install that old mac os version as a Virtual Machine (VM Fusion, Parallels, or Virtual Box). Then install FMP 11 on the VM with that older mac os and just use that VM when you need to use FMP 11. Setting up a VM is not difficult. There are tons of "walk-throughs" on the Web.
HOPE THIS HELPS.