First, make sure nothing is hidden in the module you published, especially on the time dimension side. I put this together and it works, but remember I am didn't hide anything nor do I have any filters applied.
Published it to the dashboard as well as the years (Publish Page Selectors for Dashboard) from a module that is only dimensionalized by Years.
Open the designer while clicked on the data module and select Sync
Data should change now when you toggle the different years:
Again, if your selection to sync is greyed out, it is because you are hiding something.
If the boolean "Synchronize Selection" is not active anymore for the "Time" list of the module: verify if there are filters applied to the "Time" list in module in dashboard. this could generate the disable of this option.
Another possibility: there are manually hidden time columns in the dashboard: verify if "Show ALL Columns" is active.. if yes.. then you need to Show all columns for Time and then the flag will become again available.