Same here. As the squash develop, I place a small bundle of straw under them. Soil contact seems to be the primary source of infection, since that is usually the site where spoilage begins. The fungus apparently remains inactive (probably because the squash has defenses) until the squash has weakened in storage. Harvest should be done carefully, any flaw in the skin is a potential entry point for infection. Immediately after harvest, I carefully wash down the squash to remove any clinging soil or insects, then cure them in my unheated patio.
I store winter squashes in my semi-heated basement, on ventilated shelves several feet above the floor. This is not ideal, I wish I had a less humid location... but you play with the hand you're dealt. DW would probably not approve of storing them upstairs, it is miracle enough that she has accepted the table of seed trays in the living room.

Regardless, most of the squash will last into January. The squashes will spoil on a staggered schedule, so they should be inspected frequently & used immediately at the first sign of spoilage. There have been a couple over the years that actually made it to Spring (they are gourds after all) but they were few & far between, maybe 5%... and not really palatable at that point. If I get 4 months of storage, I'm a happy camper.
There is one other storage possibility - dehydration. I've tried that with kabocha squash; peeled, sliced, then dehydrated. Store the "chips" in an air-tight container immediately after dehydration. To both re-hydrate & cook, I placed the chips in a rice cooker with a little water; the flavor & texture were similar to fresh baked, very pleasant. With a 9-shelf dehydrator, I can do about one complete squash at a time.
The only thing that has prevented me from doing this on a larger scale is the difficulty of peeling the squash; I need a squash that is easier to peel, and/or a faster method of peeling squashes with hard skin. The Buttercups I grew last year were so hard-skinned, I only peeled two before giving up. This year was supposed to be an attempt at Option A, a kabocha-like squash I've grown in the past that was easier to peel - but it perished under weed pressure.

Hopefully the same variety will fare better next year. I will still be trying dehydration with the ripe Tromboncino I am growing for seed, even though the quality of the ripe squash is IMO inferior to their Butternut cousins. They will certainly be easier to peel! Who knows, the texture may be improved, since I can control the amount of water used in re-hydration.