To some extent, I have to disagree. If they wait until it becomes an issue to make it an issue, it might be too late.
While I disagree with James about not making it an issue at all, I dont know how worried I be about it. I'm sure the FAA has seen and heard of an N number being spoofed before. You certainly wouldnt be the first and you wont be the last person for it to happen to. From the sounds of things, you and your plane are no where near Florida at the time of these flights so you can easily demonstrate you aren't there. I also would note that being alerted by FlightAware gives you early warning that someone is using your N-Number but generally speaking N-number spoofing goes undetected by the pilot until the FAA is knocking on their door so you wouldn't be the first pilot to make the claim.
As to what to do, this is what I would do:
1) Get a notebook and start recording where and when your N-Number is spotted that isn't you. Also log where you are and your plane is at the same time. This would be good evidence to present to the FAA.
2) If possible get airborne in your plane ASAP. Nothing says "it wasn't me" like your plane showing up in 2 places at once or traveling impossible distances in a short period.
3) Figure out where the plane was reported and attempt to listen to ATC over on liveatc.net. Save the recording if you can find the one with your N-number (or a similar number misheard) being used.
4) Contact the ATC control in the area where your N number is being erroneously spotted and advise them of the situation and see what they say. Include reporting it in your log everytime you do report it to the local facility.
This will provide you with a nice record that CYA's if and when the FAA comes knocking
1-4 are a lot of work, just keeping your own flights and plane log accurate takes time.
I would do 1 thing first.
Create a NASA Report it's "confidential" and has some real benefits for the type of FAA interaction you might encounter.
Also, those reports do get studied, so specifics may help in the investigation. Here is the link:
https://asrs.arc.nasa.gov/
Benefit. if the other guy busts bravo, and you can't convince the FAA it wasn't you, as long as no metal was bent, you experience a much friendlier FAA.
I have not actually used this, but if I even pass through Delta un-announced, I will do this.
In fact, if you decide to land left downwind and the chart calls for right downwind at an uncontrolled airport, you should do this. You never know who will complain.
Finally these NASA Reports do help make GA more safe. They are important to the process.
These will _not_ help you Busting a VIP TFR, as that gets reported to the DEN.