OK, the addresses are in OSM. If you look for “2004 Swan Lane, Pinellas County” you will find the street.
It does not know it is part of Palm Harbor though. This is probably the case for the other addresses as well.
Two possible reasons:
- Addresses use bad Tiger data (indicator is that there is no address node where the house is supposed to be)
- We only have a node for Palm Harbor and not an area, in which case it is pretty hard to know to which town/neigborhood/… the street belongs
Solutions for both problems: map them 
This problem with OSM mainly exists in the US (and less in Europe), because, due to lack of properly mapped addresses, Nominatim relies on Tiger data. I have no idea what the Garmin map uses here. I doubt it falls back on Tiger data. So it is very well possible the address is not in the data used by your Garmin map.