I can't help you with the green card issue, but I can with the travel questions.
Some of it depends on how old your child is. If it is school age, you may want to choose a place and settle quickly, but if not, then you have more opportunities to find a place that works for you. There is a world of difference in California and Florida, yet they are both warm! If you choose a travel company, you can check out several places before deciding. Of course, this depends on your family situation too, but you may take a 13 week assignment in several places before you make a final decision especially if you are not in a hurry.
You will find the cost of living in California to be much higher than Florida, but salaries are higher. If you are going to southern Calif (warmer), be aware of the crowded highways, outrageous home prices, etc. Florida has a broader choice in housing prices while still being warm.
My personal favorite is Arizona. You get the benefit of warmth while cost of living is not so bad. I am in Tucson now, and love it. We came here for the winter as well, and are now considering a permanent move. There is plently of work for all areas mentioned as lots of people like you (and us) want to enjoy the nice weather. We have a little farm in the Ozark mountains and went home last week for a visit and the first night the temp was 20 degrees and I could not wait to get back to Tucson!
I work for RN Network and I know they have Canadian travelers. Give them a call and ask for specifics as far as trying it. You are welcome to PM me if I can offer any info. Good luck, and welcome to the world of sunshine, shorts and sandals!