Well, one possibility could be that Zetsu imparts some unique and dark jutsu to Naruto. Maybe Naruto starts to learn stealth and subterfuge skills from Zetsu, which are different from what he learned at the ninja academy. This could lead to Naruto having an edge in certain battles where traditional ninja skills aren't enough.
There could be internal antagonists within Naruto's own village. Some of the more conservative ninja in the village might not trust Naruto's new - found skills and powers obtained from Zetsu. They could try to undermine Naruto or even attempt to exile him, fearing that he has been corrupted by Zetsu's teachings.
It's likely that Naruto's chakra control and power would develop in new and unexpected ways. Black Zetsu might introduce him to strange chakra - based abilities that are not part of the normal ninja training. This could give Naruto an edge in combat, but it also might make him a target. Since Black Zetsu has his own motives, Naruto could be dragged into some dangerous and complicated situations that he may not be fully prepared for.
He may develop a more calculating side. Black Zetsu is all about manipulation. Naruto could start thinking several steps ahead in a more devious way. For example, he might start using people's weaknesses against them, which is completely against his original nature. This change in personality would likely cause a rift between him and his friends who know him as the always - honest and straightforward Naruto.
Well, one way could be in their physical abilities. Maybe in the fanfic, Naruto is shown to have some of Zetsu's stealthy movement skills, like being able to blend into the environment in a similar way. Another similarity could be in their connection to nature. Zetsu has a strong link to the plant - like aspects of nature, and Naruto's chakra might be shown to have a similar kind of connection in the fanfic.
When Naruto absorbs Zetsu in the fanfiction, it could impact his mental state as well. Zetsu has been involved in many dark and devious plans. Naruto might have to struggle with the knowledge and influence of those. However, on the positive side, Naruto could use Zetsu's ability to communicate with the plants and gather intelligence from the natural world. This would expand his range of information sources and give him an edge in strategic planning during battles.
If Naruto has Black Zetsu in the fanfiction, it can change the power dynamics. Black Zetsu might try to use Naruto's power for its own ends. Maybe it convinces Naruto to use his power in a way that seems good at first but actually has hidden motives. This could lead to new alliances and enemies forming in the Naruto universe. For instance, some villages might see Naruto as a threat now that he's being influenced by Black Zetsu.
It's relatively common in some circles of fanfiction. Writers often like to blend different bloodlines in Naruto fanfiction to create more complex and unique characters. By giving Naruto Zetsu's bloodline, it can add an element of mystery and unpredictability to his character. For example, he could potentially have some connection to the Akatsuki in a different way than in the original story.
Yes, there are likely some out there. You can start by looking on fanfiction platforms like FanFiction.net or Archive of Our Own. Search for 'Naruto x Zetsu' and then filter through the results to find stories that are not of the 'lemon' variety. Usually, these platforms have tags and summaries that can help you identify appropriate stories.