We go to florida every year either before or after Christmas and the weather is most definitely not crap! this year it was in the mid 70's to low 80's every day. last year it was in mid to low 70's every day and really lovely to walk around in imo. how old are your dc's mine now prefer florida keys. never gets over 100 and there is loads to do for older/outdoor kids. not been to tampa yet but lots of mnetters say this is good too. can definitely get under the budget of 10K easily. Disney is really busy over Xmas so if you can go a week either side this would help with the queues we take our DC's out of school as before Xmas they don't seem to miss much and this is the only time we can go. PM if you want details of places we have stayed/visited. also look at accuweather site to tell you how warm it will be.
How about a Caribbean or Mexico family resort or a Cruise. My family had a great holiday staying in L.A for a few days and vising Disney and then cruising along the Pacific side of Mexico. We went Feb half term and California was Jeans, t shirt and cardigan weather and Mexico lovely and warm but not too hot. There is so much for the children to do on cruises.
Have you thought of Kenya ? It's really hot at Christmas time, your money goes a long way when you're there and if you go to somewhere like Hemingways in Watamu in Malindi, the deal includes Bfast, a huge "tea" and dinner, trips to local town are free as are snorkelling trips from the hotel jetty, we've been three times now and love it. Here's a link to my review of our Easter hols there from my blog www.doindubai.com http://doindubai.com/2013/04/17/deep-sea-fishing-hemingways-on-kenyas-coast/