Greetings!!! And welcome aboard!
You are at the right place. Great information here.
Although Delaware is reciprocal with North Carolina once you become a Delaware resident you may not carry on your North Carolina permit
You will need to find five citizens of your county that can attest to your character
Not really sure how this works out for a new resident of the state but the fact you already have a North Carolina permit should work in your favor.
It may be worth your while to contact the Atty. Gen.'s office to find out if there is a problem for a new resident getting a license.
My understanding is that they would like to see that these five residents have known you for a year.
This will probably be your biggest issue
Unfortunately there is no quick or cheap way to do this.
Even with references you are looking at about $400 and a six month process.
On a positive note, you will find buying a handgun easier here. You will not have to get a purchase permit from your sheriff. Just the NICS check.
You will find a lot of information and help on this forum. Maybe others will have some ideas.
Remember that once you establish residency, your reciprocal North Carolina permit will no longer be valid in Delaware
Again, welcome !!