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Once you have completed your DUI suspension period, you should have received a Notice of Restoration in the mail....and you would need to renew your license. Also the Mandatory Probation Period starts after completion of Probationary Driver Program, Driver Improvement Program, official warning issued by MVC or restoration after points.
If all of that has been completed, and your license has been restored, then you should be in position to apply for a license in FL.
But, if it hasn't then NJ is likely going to have a hold on your license and you'll have to contact the MVC and find out exactly what you need to complete in order to be able to get a license in another state.