I always doubted if I should include ID or the actual text in the dimensions. For example, see DimSalesPerson in the screenchot below - https%3A%2F%2Fwww.codeproject.com%2FArticles%2F652108%2FCreate-First-Data-WareHouse&psig=AOvVaw2oGn_eyUgi8KV9m_u6PSIF&ust=1666197880975000&source=images&cd=vfe&ved=0CA0QjhxqFwoTCOjG9_6c6voCFQAAAAAdAAAAABBJ. Instead of using StoreName as text in DimSalesPerson, they used StoreID. What do you usually do in the cases like that when there is an attribute in a dimensions table A that has a stand-alone dimensions table B?
Dimensions should never link to other dimensions. The only exception that I normally make to this rule would be for dimensions with dates that would join to a date dimension. I'm sure there are other exceptions, but they're rare and I don't think they apply here.
So yes, the correct answer, based on what you provided, would be to recreate some of the DimStores dimension attributes inside of the DimSalesPersons dimension, based on the store that the salesperson is associated with. Depending on the type of dimension that DimStores is, your ETL processes will need to handle changes to an existing store record in both DimStores and DimSalesPersons. That's a pretty common design paradigm.