If you refuse the demotion, and they say "Then you are discharged, quit, are fired, need to go home!"..whatever, you should file immediately after your last day of work for unemployment benefits. They aren't much, but they are there for you when you are out of work through no fault of your own. When you are offered a significantly less good job than you have had, and that is your only choice, then unemployment insurance is your only option if you do not want to accept the new job. As cbg noted, they are not obligated to provide you with anything except COBRA.
If you accept the new job and work at it, even for one minute, and then quit because of the demotion ("I can't live on what I'm making!") then you have just about a zero chance of being approved for unemployment. You've already accepted the new job, pay rate and all.
Since you live in Florida, where the maximum unemployment weekly benefit is very low ($275) regardless of how high your salary was on the job you were working at, and they've made unemployment a very difficult program to get signed up on anyway, I would advise you to think carefully about quitting. Even a fraction of your former wages might be more than you could draw in unemployment, and while you are still working there, you could search diligently for something else that pays better.