Here are the ACT dates for the spring: http://www.actstudent.org/regist/dates.html#first .
I would shoot for the June and the September 2016 dates. Both dates should give you enough time for the deadlines, though there were some issues with ACT being a little sluggish releasing September and October scores this year.
Also, have you considered trying the SAT? Not saying you should, just wondering if you have done it yet. They are changing the format so be aware of that.
The main thing is to keep an eye on all the deadlines - testing and application-wise. Consider making a spreadsheet of everywhere you’re interested in going, and the pertinent dates and items needed.
FSU also has an essay, and a suggested resume they want you to submit. So in the summer you could polish up (or create from scratch LOL) those items too.