If you have never gotten ripened figs then you may have a tree that requires caprifcation, aka to fertilize the tree. This process is induced by a a tiny "fig wasp" that only exists in Cali and Mediterranean climates. The fix to his problem is to use the tree to graft a common fig that does not need the wasp, or plant another tree.

No, you cannot pick figs and sun ripen them. They dont ripen like a tomato of the plant. Instead, the fig might just spoil or dehydrate.

Let us know if you have any more problems.

Katrinanixon, the short answer is if the figs are not ripe on the tree, picking them and placing them in the sun to ripen is not a viable option. As I understand it, in the nine years that you lived in your home you haven't ever eaten ripe figs. If that is correct it could be that the envirnomental conditions are not ideal but it is more likely that the tree requires caprifcation or the help of a specific wasp that is not indigenous to NC. Sorry for the bad news but now at least you can decide what to do with the tree.

If the "tree produces big figs" it doesn't seem likely that lack of caprification is the problem. It's my understanding that unpollinated figs that require pollination will drop - not get big.
I'd guess that the shade is the problem. If you can't do anything about that, maybe pruning the tree or thinning the fruit would help. It can't ripen its crop now, but perhaps it could if there were fewer figs to ripen - some is better than none.

Thinning the tree itself,and leaving only 3 trunks or less will make the sun penetrate the canopy better and ripe the fruits.
By thinning also the number of fruits will be much less so they will grow and mature much faster.
It sounds like you have a tree that get ripe late,and require a lot of heat units to produce ripe fruits.
Only by having less canopy ,and fruits one can hurry the maturation of the fruits.
If that do not work,then replace it with an early cultivar,as Hardy Chicago,Celeste,Tacoma Violet,Marseilles vs black,Gino's fig,Etc.
These are getting ripe year after year here in New Jersey,where the climate is colder than in your climate.
It is all in the genes mainly.
Happy Gardening.
H