I believe the answer you seek will vary based on the particlar question related to dual residency, e.g., for tax purposes, firearms purposes, voting purposes, etc.
In general, one is a resident of a state where they are physically present without any current intent to leave. Residency is manifested by objective factors including the state you are phsysically present in, where you vote (a big factor), where you pay state taxes, where you own property, where your vehicles are registered, etc.
BATFE has said the following (note they use the term reside which is not necessarily the same as being a resident of the state):
Q: What constitutes residency in a State?
The State of residence is the State in which an individual is present; the individual also must have an intention of making a home in that State. A member of the Armed Forces on active duty is a resident of the State in which his or her permanent duty station is located. If a member of the Armed Forces maintains a home in one State and the member’s permanent duty station is in a nearby State to which he or she commutes each day, then the member has two States of residence and may purchase a firearm in either the State where the duty station is located or the State where the home is maintained. An alien who is legally in the United States is considered to be a resident of a State only if the alien is residing in that State and has resided in that State continuously for a period of at least 90 days prior to the date of sale of the firearm. See also Item 5, “Sales to Aliens in the United States,” in the General Information section of this publication.
[18 U.S.C. 921(b), 922(a) (3), and 922(b)(3), 27 CFR 478.11]
Q: May a person (who is not an alien) who resides in one State and owns property in another State purchase a handgun in either State?
If a person maintains a home in 2 States and resides in both States for certain periods of the year, he or she may, during the period of time the person actually resides in a particular State, purchase a handgun in that State. However, simply owning property in another State does not qualify the person to purchase a handgun in that State.
[27 CFR 478.11]
Examples are present in 27 CFR 478.11 (weird huh):
State of residence. The State in which an individual resides. An individual resides in a State if he or she is present in a State with the intention of making a home in that State. If an individual is on active duty as a member of the Armed Forces, the individual's State of residence is the State in which his or her permanent duty station is located, as stated in 18 U.S.C. 921(b). The following are examples that illustrate this definition:
Example 1. A maintains a home in State X. A travels to State Y on a hunting, fishing, business, or other type of trip. A does not become a resident of State Y by reason of such trip.
Example 2. A maintains a home in State X and a home in State Y. A resides in State X except for weekends or the summer months of the year and in State Y for the weekends or the summer months of the year. During the time that A actually resides in State X, A is a resident of State X, and during the time that A actually resides in State Y, A is a resident of State Y.
Example 3. A, an alien, travels to the United States on a three-week vacation to State X. A does not have a state of residence in State X because A does not have the intention of making a home in State X while on vacation. This is true regardless of the length of the vacation.
Example 4. A, an alien, travels to the United States to work for three years in State X. A rents a home in State X, moves his personal possessions into the home, and his family resides with him in the home. A intends to reside in State X during the 3-year period of his employment. A is a resident of State X.
Assuming you lived 182.625 days in one state and 182.625 day in state 2, then where are you considered a permanent resident? In many instances, a court will look to where you vote, where you have a driver's license, where your cars are registered, etc. as the state of permanent residence. To my knowledge, no state will give you a driver's license generally any longer without taking the one you have from your previous state - some states used to issue a second driver's license valid in that state only, however (FL used to do this - call the FL DMV and see if it still happens). Unsure if this practice still exists.
So, in sum, you can only be considered a permanent resident of one state but you can reside in a second state, based on manifestation of objective factors and owning property, for purposes of purchasing firearms.
All that being said, I would surmise that some FFL's might be weary of transferring the handgun if you provide an out of state DL. If the state only FL driver's license still exists, then once your father gets one of those, no one will ask any questions.