If you are a Delaware citizen then a Florida non-resident permit is not valid in Delaware, which is why I asked about your home of record. Do you vote with an absentee ballot as a Delaware resident? Delaware will only honor permits from other states if you are not a citizen of Delaware, at least which is how I interpret the code. I could be wrong but read this and see what you think.
TITLE 11
Crimes and Criminal Procedure
Delaware Criminal Code
CHAPTER 5. SPECIFIC OFFENSES
Subchapter VII. Offenses Against Public Health, Order and Decency
§ 1441. License to carry concealed deadly weapons.
(j)Notwithstanding any other provision of this Code to the contrary, the State of Delaware shall give full faith and credit and shall otherwise honor and give full force and effect to all licenses/permits issued to the citizens of other states where those issuing states also give full faith and credit and otherwise honor the licenses issued by the State of Delaware pursuant to this section and where those licenses/permits are issued by authority pursuant to state law and which afford a reasonably similar degree of protection as is provided by licensure in Delaware.
