汚れ and 染み: dirt and remaining stains
Distinguish a broad dirt noun from a particular stain in context.
A broad term for dirt
汚れ (よごれ) means dirt or soiling. 「靴の汚れを落としました」 says dirt was removed from the shoes.
「汚れが目立ちます」 says the dirt stands out. This is not a numerical evaluation of its amount.
A stain on a particular area
染み (しみ) can refer to a stain left by liquid or another substance. 「シャツに茶色い染みがあります」 locates a brown stain on a shirt.
「この染みはコーヒーで付きました」 states that coffee caused this stain. Unlike guessing from color, the cause is explicit.
Checking the removal result
「洗いましたが、染みは残っています」 says the item was washed but the stain remains. Separate the washing action from successful stain removal.
「汚れを全部落とせたわけではありません」 says not all the dirt could be removed. It does not establish that none was removed.
Does 「洗いましたが、染みは残っています」 say no washing occurred?
Show the answer and why
No. Washing occurred, but the stain remained.