I stayed on Koh Phayam for 2 weeks in Jan. Sent an e-mail to the guesthouse a few days before. Xmas is busier, but depending on your budget you can always find a place. Do you know which side you want to stay on? The quiet side, nicer beach. Or the side with more nightlife and facilities. You'll need a scooter to go from one side to the other.
You can reserve at most places without having to pay in advance. So might as well.
P.S. Keep in mind there are no banks or ATM's on these islands, so bring all the baht you think you'll need. Electricity is not 24hrs at a lot places, generator from 18:00 - 23:00 - Not all places have Wi-fi.
little Koh Chang seems to attract a lot of Germans for some reason and is more rustic than KP. Nothing wrong with that
Also the beach is not as nice I hear - That's why I took a pass.
I'm often on Ko Phayam around Christmas time (not last year though) and there's usually a huge influx of visitors from about December 20th onwards. It's not unusual to see people sleeping in tents in the grounds of the more popular resorts waiting for a room to become vacant.
I'm not so sure you'd be bound to find a room during the week between Christmas and New Year .... except maybe at one of the more expensive upmarket resorts like Blue Sky Resort or Payam Cottages.
So yes, I'd advise you to book if you're going then, especially if there's a particular resort you want to stay at.
I've never stayed on little Ko Chang either but it does seem to be a bit of a German enclave...everyone I meet who stays there seems to be a German speaker anyway!
Not sure about Khao Sok, have never been there during peak season but I'd guess you'd be OK. There are lots of places to stay and most people seem to head for the beach around Christmas time.