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Yes, you an do it this way if you like. I typically keep mine in brown paper bags in a dark location. This article will help as well: https://www.gardeningknowhow.com/ornamental/flowers/fgen/harvesting-garden-seeds.htm

You can cut the plant down to about 6 inches front he ground level as soon as the blooms are spent.
This plant can tower to 6 feet in height and yes, will spread fairly rapidly.
You can lift the plant really at any time now through fall, or wait until spring. I like to move and split my plants in the fall, as sometimes it grows so quickly in the spring it is difficult to get to it before it gets to large.
Hyssop is loved by the bees, and though they are a necessary part of gardening for pollination, they can be not as welcome while your sitting and dining. Moving your plant to another area may be the best solution.

I do find that the butterflies enjoy my Hyssop. The Monarchs and Swallowtail have been enjoying the last blossoms for the last couple of weeks.

Here are a few links for you with more information.

https://www.gardeningknowhow.com/edible/herbs/hyssop/growing-agastache.htm
https://www.gardeningknowhow.com/edible/herbs/hgen/flowering-herbs-for-bees.htm