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Store your plants in a cool, dark place and they should be fine. If the plants look like they're drying out, sprinkle with a little water. Keep in mind that a bit on the dry side is better than too wet, as excess moisture may cause rot.

It would be best to transplant during dormancy.
This article will help you.

https://www.gardeningknowhow.com/edible/vegetables/asparagus/transplanting-asparagus-plants-tips-for-how-to-transplant-asparagus.htm

What a wonderful sounding bed of Asparagus.
Since it is so well established, it likely is spreading out to gain more room.
You may want to lift some of the crowns and either divide and replant or even share with another gardener.
You can cut the seeds from the plants to eliminate reseeding.

https://www.gardeningknowhow.com/edible/vegetables/asparagus/growing-asparagus.htm

Heavy clay soil can kill our the plants.
These links will refresh you on the care requirements.

https://www.gardeningknowhow.com/edible/vegetables/asparagus/growing-asparagus.htm
https://www.gardeningknowhow.com/edible/vegetables/asparagus/planting-asparagus-how-to-make-an-asparagus-bed.htm
https://www.gardeningknowhow.com/edible/vegetables/asparagus/transplanting-asparagus-plants-tips-for-how-to-transplant-asparagus.htm

This article will help
http://www.gardeningknowhow.com/edible/vegetables/asparagus/cutting-asparagus-foliage-back-in-autumn.htm

Let them flower and seed. You can actually plant the seeds next year or throw them out. They are very toxic so be careful. Don't trim them back til you move them. This will make a better harvest the next year. I don't know if they are in pots or a garden bed but if your digging them up you have to dig a very wide deep hole since the roots spread out flat and deep. Resow the roots on a mound and drape them over it.