I will recommend you some and I'll give you short videos to illustrate it. I hope it will be helpful. However, I have not watched a lot of anime, so please take that into account.
My favourite was LoGH. I really loved the characters and the scope of the story. I would recommend it to anyone. Some people say it's a bit slow and not that good in the beginning, but I liked all of it.
There's been a remake last year that covers the start of the story, and I liked it, but it somewhat lacks the calm charm of the original.https://www.youtube.com/watch?v=QFKXrtjNw9Y
I liked Cowboy Bebop a lot, as you did, so I'd also recommend Samurai Champloo, which was made by the same studio. It has a lot of style, and a really 'cool' feeling to it. I also liked the cultural fusion of medieval and modern Japan.https://www.youtube.com/watch?v=97Ls7tbmbyk
I have only read Mushishi, and not all of it, but I liked it a lot. It is an episodic anime, with the travelling "doctor" solving a different problem in a different place each episode. However, what made me like it is the depiction of nature as alive and filled with spirits. It is done beautifully, and it has a very peaceful and calming feel to it. Some episodes also explore characters in an interesting way.
I haven't found any good clips, I'm sorry.
I've read Monster a few years ago. It's about a Japanese neurosurgeon in Germany. I thought it was very good. It is really emotional.
This scene is not a good choice, but I couldn't think of another that was illustrative and not spoilery. It's not ideally illustrative, and it mentions an important plot point, although it doesn't spoil it.