Pink Banana squash

I am not sure when to pick my pink banana squash. The vines are struggling from age, disease and drought. The picture shows a nice big squash. My question is can I cut them off the vine now? Or is there an advantage, or at least no disadvantage, to leaving them on their vines.
I have been hand watering and the vines have perked and started putting out new growth which makes me think the squash may be benefiting still being attached. Its still hot here in Georgia, in the 90s, so it isn't close to any frost.
Any suggestions on what to do with all the squash? I have about 5 huge ones. They are approximately 18-24 inches long.
Thanks for your input
Helen

Here is another squash showing more vine. It grew very impressively all over the bushes and out onto the lawn.
I would like to start cleaning it all up but will wait if the fruit will be better if I wait.
Helen

Since it's a winter squash wait until the stem end is completely brown and then cut it. Be sure to leave several inches of stem attached. That will facilitate longer storage. I have found that winter squash stored outside the refrigerator lasts much longer than those that are refrigerated. Wait at least another four to six weeks after picking to cook winter squashes. The added curing time intensifies the flavor and sugar content. I've not grown this particular squash but would think you'd use it any way appropriate to winter squash...peel and pan saute, halve and seed then oil flesh and roast with flesh side down, steam and mash with butter and herbs, make pies, soup, etc..

Thanks Laurel. You told me just what I wanted to know. I will leave them alone.
When you say they are better left out of the refrigerator where do you suggest is the best place to store them. Outside in a dry place or in a cool place inside, like say the basement?
Thanks again.
Helen

I use them as centerpieces on the breakfast room table or the kitchen bar counter. Here they are in a big basket at our cottage (Maypop). I have a huge platter with squashes in Atlanta. Some varieties hold better than others. Butternuts are really good for storing as are long pie. I had some that went from last fall until this July just sitting on the counter. Delicata (the striped ones) only hold for about three months. They seem to do best where it is relatively dry and there is good air circulation. Temperature does not seem critical. A damp refrigerator drawer is the kiss of death though.