Yes you can keep your permanent residence in Florida.  The income that you earn in North Carolina will be taxable

to North Carolina.  Since you are a resident of Florida you will file your North Carolina return as a non-resident to claim

only the income earned in North Carolina.

You are required to file in North Carolina if your income from North Carolina exceeds

Single $10,000

Married Filing Joint $20,000

If you need to file in North Carolina the form you need to file is  D-400 Schedule PN 

@DianeC958  _ I read the post differently.  @raymndmark  is currently an NC resident and wants to remain an NC resident.  He can do that and, in fact, MUST do that, since his time in FL is expected to be temporary (less than 1 year) and he maintains a permanent NC abode.  He will continue to file a resident NC tax return.

The usual issue of how to deal with taxes, in the temporary resident state, is moot since FL does not have an income tax.