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You should be able to do either or both ... you can transplant or plant seeds. I don't know much about the seeds from Spiderwort but I have dug up plants and moved them to other areas of the garden or put them in pots. We have it growing and spreading through our lawn. It is very hard to dig up from thick grass.
Here's the link to Plant Files for the different types of Tradescantia (Spiderwort): [HYPERLINK@davesgarden.com] Scroll down and see if you spot one that matches yours and if you click on it, you will see more photo's and find growing and care information.