LadyinRed

OK so I am taking a large chunk of time off work to go travelling with my 7 y/o up north (from Phuket). A lot to do in the first week in and around CMX but I have not had anything planned after our stay in Chiang Dao; so from 25th Dec onwards we'll be 'on our own' - no itinerary, no accommodations booked (till 31st Dec in Chiang Mai). I'm thinking of going to Pai from Chiang Dao before making our way back to CMX in time for new year's eve, but I'm not sure now, so I'm asking:

1. Will it be hard to find walk-in accommodation on Christmas day in Pai? We're not fussy, 500 THB/night is fine but can also afford up to 2k

2. Another option is to go to Chiang Rai instead of Pai, is this a better option?

3. If I do make it to Chiang Rai is it worth travelling to Nan before heading back to CMX?

I have a week from Christmas day to new year's eve, so time is not really an issue. We're either on rental motorbike or local buses, and just want to experience something different from Phuket for my kid, and if you know these areas well, why/where/what would you do?

Thanx in advance