My first year growing and curious as well. They are doing so well in my climate i want to continue every year. A good climber, lots of fruit. (or call it veg). I've pickled it, roasted with fresh corn and tomato, basil. , etc.
It is a gourd so i think it may do well by leaving a few to harvest with the winter squash? (For saving seed?)

I don't have experience with this type, but I think the method would be the same. Let it ripen fully, cut it open, pick seeds out, place them in a single layer on parchment paper, place in a cool dry place for about a week, bag it up and freeze it (paper is best, but I just use plastic bags) and take it out and plant next spring! That's what I do, unless there's something I don't know about this variety.

Is getting a hard shell while on the vine the indication that they've "fully ripened"?

Because mine are NOT gourd-like hard yet.

Let the fruit harden on the vine. Don't have experience with this particular gourd. But with squash and pumpkins it is ALWAYS much preferred to let them ripen on the vine as much as possible. Maturing seed requires much energy and the plant supplies this.

George
Tahlequah, OK

Remember that Cucuzzi is a long skinny bottle gourd (Lagenaria siceraria) not a squash. There are several bottle gourds used as summer squash substitutes. Opo is probably the more popular. Seed saving is exactly the same as if you were growing it as a craft gourd. let dry completely at which time it will be hollow and the seeds loose in the shell. The dried form can be used in crafts as a snake gourd.