That sounds rough. Firewood containing a lot of moisture will produce more smoke. It might help to burn older branches and stumps containing less water. If the chimney has leaks, they might be plugged with some type of concrete and sand mixture? I'm not certain what the approved solution is but those types of conditions can't be allowed as they are actually dangerous.
Yes!! Thanks for your care and attention in this great reply, dear @sanity-check ! 🙏💝😊 The chimney we had problems with was in our rental home, so it us for them to resolve, thankfully; we have a small but super-efficient wee stove g chimney in our crazy house on our land, and have a good rotation of mini logs for it - aye; we are well-versed in good stove management (me being from a mountainside on a Scottish island), and are loving being in a place where everything is (ironically) much safer and more peaceful 💝🙏