Anything you found on a records search will be found by others who check up on you, so you were smart to check on yourself, first.
The felony charge you describe was dismissed and not prosecuted. Therefore, it was not a conviction or adjudication withheld, and you do not have to disclose it on your job application.
However, you still have the problem of the fact that a record check will show that you were arrested and initially charged. Be prepared to give a clear explanation of what happened because it's reasonable to think that a potential employer will ask you about this.
The second case - the DUII - involved a plea of "no contest." Therefore, you must disclose it based on how the question on the job application is worded.
I urge you to immediately contact a licensed Florida attorney to ask the following questions:
1. Can your DUI case be expunged? In many states, DUI charges cannot be expunged, by law. Find out the Florida law.
2. Can your record of arrest for the felony charges be expunged?
Be aware: Many job applications, especially for government jobs, include language on an application stating that you swear to the truthfulness of your answers. If you misrepsent your record and your misrepresentation is discovered, you could face new criminal charges.
Your Florida DUI cannot be sealed or expunged by statute. Nor can your felony arrest be expunged since you have a withhold for reckless. The reckless, and the record of your felony arrest, may be sealed, however. Sealing preserves your record but hides it from everyone but government agencies, licensing boards, and others permitted to see sealed records. A sealed record MAY even be visible to a university, but I am not sure.
As for the application, you must disclose the DUI arrest and the plea of no contest to the reduced charge of reckless. Reckless driving is punishable by up to 90 days in jail making it a hybrid charge somewhere between a 1st and 2nd degree misdemeanor. Be safe, and disclose. Just because you can find no record of this charge does not mean others will have the same problem.
You don't have to disclose the felony case, it was dropped. You don't have to disclose the reckless driving conviction either, it is a second degree misdemeanor. Even though you were originally arrested and charged with more serious offenses, the results of those cases is what controls your answer on the application. You had some excellent results on your cases and you should be extremely thankful to whomever your attorney was.