So I'm in Adachi's dungeon on 12/08 and I know the deadline is 12/24.  If I beat his dungeon today, can I still play to the 24th or does the game just end?

you will jump to 12/25 for a special event, after that you will jump to spring where you get a shot at the true ending.

No, there's a little more afterwards. I think there's also a time jump once you've finished that dungeon, though, so you might want to stall as long as you can just to take advantage of the extra days in December for last minute grinding/social linking before the point of no return.

" No, there's a little more afterwards. There's also a time jump once you've finished that dungeon, though, so you might want to stall as long as you can just to take advantage of the extra days in December before the point of no return. "

The day you want to finish is 12/22.  After that and you'll miss a special event, before that and you'll lose SL opportunities.

If you really don't care about the Christmas eve event, with whichever girlfriend you invite over, the 23rd is actually a better date for being inside the TV. The fusion forecast is "Bonus S. Link Exp" AND "Bonus Stats" with NO conditions.
 
But ultimately it's probably better to beat it on the 22nd for the little event. Popularity leads to intimacy.