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When the plants turn brown, you can cut them back to the ground. You can then add 4 to 6 inches of mulch such as compost, leaves, pine needles, or straw, making sure any mulch you use isn't full of weed seeds. Please see the last paragraph of this article:
https://extension.umaine.edu/gardening/master-gardeners/benefit/grow-it-right/asparagus/asparagus-faq/

Wait until the ferns turn brown by themselves, then cut them down. It's also important to apply compost, straw, pine needles, or other mulch over the bed to protect the plants from the cold, as explained here:
https://www.gardeningknowhow.com/edible/vegetables/asparagus/cutting-asparagus-foliage-back-in-autumn.htm

In warm regions, asparagus should be planted in winter, while in colder regions, it should be planted in early spring as soon as the ground can be worked. If you are in a warm region, see this guide:

https://pender.ces.ncsu.edu/2012/01/start-an-asparagus-patch/

Cutworms are a possibility- the following sites provide photos of cutworm damage that you can compare to your plants:

http://msue.anr.msu.edu/news/a_tale_of_two_cutworms_cutworm_control_in_asparagus

The milky white substance could potentially be a slug or snail trail. Slugs do eat asparagus, so that's a possibility too. The beet armyworm caterpillar can also damage asparagus spears. Checking all over the asparagus patch at multiple times of day might allow you to catch the culprit in the act.

https://extension.umd.edu/growit/insects/slugs
http://ipm.ucanr.edu/PMG/r7300411.html
http://www.extension.umn.edu/garden/insects/find/cutworms-in-home-gardens/
https://www.gardeningknowhow.com/plant-problems/pests/insects/kill-garden-slugs.htm

I would not advise trying to lift your Asparagus, the despise being moved, as they have long tap roots.
You can top dress the plants each spring with 2 to 3 inches of composted manure and then add a 2 inch layer of organic mulch.

https://www.gardeningknowhow.com/garden-how-to/mulch/best-mulch-for-garden.htm
https://www.gardeningknowhow.com/edible/vegetables/asparagus/growing-asparagus.htm

Sawdust will lower the nitrogen content of soil since it is almost pure carbon, so it's not so good for mixing into the soil. It can, however, be used as mulch on asparagus.