Hi,
How did you configure your product to only have the choice 1 in the quantity dropdown ?
Could you provide a screenshot of the settings of the product and one of the variant so that we can better understand the situation ?
And regarding the dropdown when the items are sold out, that's because you still have the "add to wishlist" button which needs it.
But please note that if you only want the choice "1" in it, I would recommend to change the "quantity input layout" setting of the product to "none". That way, the quantity input won't appear and the customer will only be able to add the product one by one.