I'll recommend my all time favorite Star Trek novel (which I do whenever I get the chance) which is
Chain of Attack by the late Gene DeWeese. It's a bit of an oldie, from 1987 IIRC. But it's one of those novels I remember as if I read it yesterday. And it's mostly a ship based story. It's a bit of a dark horse...I don't see too many other people talking about it so maybe my opinion of it is unique (though as I always say, I am a bit of an oddball Trekkie, The Motion Picture is my favorite Star Trek film....for real....and I liked Nemesis

).
The basic idea is the Enterprise is hurled to another galaxy where they find planet after planet that's been devastated by a horrific interstellar war, in many cases centuries before. When they eventually encounter the warring civilizations Captain Kirk has to rely on some of his diplomatic skills to get them to even talk. All the while the
Enterprise has to avoid being destroyed and at the same time try to find a way back home.
I loved it because it took place far from the Federation, so there was no back-up coming and they couldn't even call for back-up. I also like the sense of foreboding as the novel starts and they discover all the destroyed planets. Then later Kirk has to use diplomacy to open a dialogue.
I love the Destiny trilogy as well but that's likely outside your parameters as it's primarily a 24th century story. On another thread someone was posting about the Furies crossover event, I liked the first book in that series (it was called
Invasion) which was an original series book.
Of course others have mentioned Greg Cox's novels. He does a really good job capturing the spirit of the original series. I really loved his 3rd Khan book, another personal favorite (
To Reign in Hell)---he's probably actually sick of me complementing that book I do it so much, but it connects the events of "Space Seed" and TWOK. That may be outside what you're looking for, though you noted you liked "Space Seed" so maybe not. It is the 3rd book of his Khan series (the first two are
The Eugenics Wars) --which were great reads as well, but you could read the 3rd book without the first two if you wanted. It was one of those novels I couldn't put down, honestly.
I loved
Ex Machina by Christopher Bennett. It's a post-TMP novel but it's a follow up to "For the World is Hollow and I Have Touched the Sky" and he too does a good job capturing the crew of the Enterprise in the appropriate period. It felt like a post-TMP, pre-TWOK story if that makes sense.
But
Chain of Attack is my #1 recommendation. It's mostly a ship based story, focused a lot on the Enterprise characters, and it includes some of what the original series wanted to be about, exploration, co-existence, finding peace.