Leaving aside the partisan aspect - you have to acknowledge that there is a difference between a statement that "The DOE is shutdown because X" and an email that says "I am unable to work because X". You made a statement before "organizations have entire departments intended for this" - yes, and those individuals are quotes as spokespeople, "A spokesperson for the department said xyz" not "I want to tell you xyz".

There is a huge difference, but both are compelled speech. The difference is that when it is a message in support of one political party the government is violating the Hatch Act (as they clearly are in this case), not the first amendment.