Congratulations on your "surprise gourd".
Is there only one gourd growing on your vine? You may have more than one, they might be hidden by the leaves.

In answer to your questions, you should leave it on the vine until the stem is brown, hard and dry. That should be much later in your growing season. Here in zone 6a, that would be late October.
As for the mold........that's what their supposed to do as part of the drying process. Once the gourd is completely dry, it will feel hollow and you will hear the seeds rattle. The drying process could take several months. After it is dry, you can clean the outer layers of moldy skin off and have a beautiful dried gourd.

If you choose to use it as a fall decoration as a green gourd, just cut it from the vine when you are ready to use it. It will still mold after a few weeks and might even dry properly.

Look through some of the postings here at the gourd forum, you might even want to grow more gourds....they are addictive!

Jan