I am on drug court in Fort Wayne, Indiana. How do I legally transfer to Pinellas County in Florida?

You contact both courts, and explain what you want. Your parole officer would be the starting point.

First is there even a drug court where you want to go? Many places do not have them. You will have to get permission from your current location just to move. Permission of the new court ( if they will take you)

Expect this to be a longer process. Also, how has been your results, if you have had trouble with court they may not allow

What is it that is pending in "drug court" in Indiana? A criminal prosecution? Chances are that no, you cannot transfer such a case to another state. On the other hand, if it is probation you want to transfer, you might, as has been suggested, ask your parole officer.

I highly doubt you can transfer your drug case and will be compelled to appear on the date they set.

Have you posted bail in Indiana? Do the conditions of bail allow you to leave the state? If not, you are in additional trouble for having violated your bail.

I believe you misunderstand, drug court is not bail, or trails, drug court is a alternative program similar to probation. The case is sent to a special court where they report to the court every week or two week, or maybe once a month.
The difference a probation officer can not order changes, here the judge can send them to jail for a few days, order new programs and so on.

So think of drug court as a probation system where the probation officer is a judge and can bypass all of the probation issues.

I believe you misunderstand, drug court is not bail, or trails (sic), drug court is a alternative program similar to probation. The case is sent to a special court ...

I don't know if you are addressing me or the OP. But as with any other sort of criminal court proceeding, I was reminding OP to be sure to consider the bail conditions, if any.

I was doing a correction since bail does not apply to drug court, drug court is a specialized alternative financed often by grants from federal government as a option to normal sentencing. So if they are in a "drug court" program, they were already in regular court and the case was sent to drug court for processing. At this point treatment and rules of behavior (like probation) is issued.