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You may be able to just scatter the Sedum cuttings across the planting area with success.

https://www.gardeningknowhow.com/ornamental/cacti-succulents/sedum/sedums-plants.htm
What a great garden shed with so much character!

Remove small infestations manually by digging up the roots.

For larger infestations you will likely need to use a herbicide that contains 2 4-D.
Follow all packaging directions.
If you use a herbicide that contains glyphoste, it will kill everything around it, as it is not a selective herbicide.

You may need to add a surfactant to the soluction. A surfactant is recommended for killing weeds with a shiny or waxy surface.
Repeat the application in a week if the plants still appear green.

I would recommend that you contact your local garden center and make an inquiry about seeds/plants - if they do not have them in stock, I am sure they can special order them for you. There are places that sell them on-line as well - a Google search will yield several offerings such as those listed below:

http://www.greatgardenplants.com/perennials/drought-resistant-plants/creeping-thyme/woolly-thyme

https://www.richters.com/Web_store/web_store.cgi?product=X6472&show=&prodclass=Herb_and_Vegetable_Plants

http://www.americanmeadows.com/woolly-thyme?

For more information on the care of woolly thyme, please visit the following link:
https://www.gardeningknowhow.com/edible/herbs/thyme/woolly-thyme-ground-cover.htm

How your Sedum transitions into late summer and autumn will vary with the varieties that you are growing.

Some low growing or creeping varieties bloom early in the season and the flowers do dry up. The plants can be dead headed to maintain a neater appearance and promote a 2nd flowering.

Some varieties such as Autumn Joy will darken and come into brilliant color as the weather cools into Fall.
Check the variety that you are growing and it's growing pattern.
If you would like to upload and image for us to help you identify your plant, please do so!

https://www.gardeningknowhow.com/ornamental/cacti-succulents/sedum/sedums-plants.htm
https://www.gardeningknowhow.com/ornamental/cacti-succulents/sedum/sedum-plant-pruning.htm
https://www.gardeningknowhow.com/ornamental/flowers/stonecrop/stonecrop-plant.htm

I prefer to cut down the Sedum in the Fall. It generally tends to fall open and is easier to clean up now instead of spring.

Sedums survive in nearly any soil conditions.

You may have planted the commonly found Autumn Joy Sedum.

An Autumn Joy Sedum does best in moist soil with a ph between 6.0 to 7.5 which would be moderately acidic.

Each spring you can sprinkle a balanced light fertilizer in granular form around your Sedum plants.

Scratch into the soil and water in.