Thanks for contacting Syncfusion support.
We have checked with your code snippet and tried with your code example. ClientSideOnChange event triggers properly when select the items in the dropdownlist and correspondingly datepicker show(), hide() method works. For your convenience, we have attached the sample in the below location.
If still you face any difficulties, please check,
#1) whether any script error thrown in console window.
#2) whether ClientSideOnChange event triggers properly.
#3) if triggers, check with the argument value (args.value), whether it is a string or integer type and changed the code accordingly.
Please revert back the sample to reproduce the issue at our end, if still problem persists.
Francis Paul A