To give an idea of how big these departments are, here's the first post-Stevenson Plan apportionment for the Electoral College.
Under the original system, the number of electors given to each province was based on a regional quota system similar to the OTL Canadian Senate, but the Stevenson Plan changes this so that each department is guaranteed one elector and the rest are distributed by population. The electors are chosen by bloc vote, and unlike the National Assembly, the franchise is still restricted to landowning men. This means the electorate is about half of adult males, disproportionately rural and just about entirely white (though it's not an explicitly racist franchise anymore).
The way the Electoral College works sort of resembles the
sénat conservateur of Imperial France, or at least there's not really a closer OTL analogy. The electors are expected to be on call in New York for most of the year, and when a senator dies or resigns, the electors from his (or her, in very theoretical theory) home department nominate a successor with the advice and consent of the general council. The proposed successor gets voted on by all the electors, requiring a two-thirds majority to be approved, and if this is not reached, a new candidate must be nominated. The process is repeated until a two-thirds majority is found, at which point the candidate approved is seated as a Senator in the subsequent session. There are about a hundred senators at any given time, quite a few of them very old, so electoring is more time-consuming work than you'd expect from a short job description.
The Supreme Governor is chosen in a similar way, except that instead of being nominated by departmental electors, he's (again, there's nothing explicitly saying a woman can't serve, but hahahahahahahahahahahahaha oh God have you
seen the 19th century) nominated by the National Assembly. Any candidate who gets signatures from more than one-fifth of the Assembly is nominated and proceeds to the Electoral College, which, as before, needs to approve a candidate by a two-thirds majority.
As you can tell, this isn't a very democratic or efficient way to run a government (have I mentioned that senators and the Supreme Governor serve for life?). Which is what happens when you mix Alexander Hamilton's plan of government with state governments fighting tooth-and-nail for what few powers they have left.