Your understanding is correct.
The question is about whether 〜を疑う and 〜と疑う mean the same thing, and the answer is they don’t.
The first sentence from the novel matches the first pattern, and this 疑う means “to doubt”. In this particular sentence, the whole of 私があの鞘の持ち主であること is the direct object of 疑う. They doubted that the speaker (私) was the owner of the scabbard, meaning they didn’t think she was the owner.
The second sentence from the manga matches the second pattern, which is similar in construction, and meaning, to 〜と思う, 〜と考える, etc. This 疑う would be better translated as “to suspect” than “to doubt”. They suspected that the speaker (私) was the owner of the scabbard, meaning they did think she was the owner although they were not sure if that was really the case.
I have no idea why such a mix-up happened when the novel was adapted to the manga.
[EDIT]
Though I generalized the question in my explanation above, the particles are not the only difference in these sentences. There are other differences that come with it.
私があの鞘の持ち主であること refers to the speaker (私) being the owner of the scabbard in general terms. It doesn’t have to be a truth. It could as well be a possibility. When it’s the object of 疑う, it means you doubt its validity.
私があの鞘の持ち主だ, on the other hand, is an affirmation that the speaker is the owner of the scabbard, although the degree of its certainty varies depending on what verb follows. 疑う adds a sense of suspicion but it’s not much different from 思う in this case.
[EDIT 2]
I have been reminded in the comments that 〜を疑う could also mean “to suspect.” In this usage, the object usually refers to something negative, such as disease or criminality. These days you can expect to get many search results containing 感染を疑う.
However, this doesn’t seem to apply in the current context at all. If I understand the description of the story correctly, the speaker is the real King Author, and therefore, the legitimate owner of the scabbard. And she knows it herself. If she says 私があの鞘の持ち主であること, that’s a fact for her. She wouldn’t be annoyed if someone suspected it to be true. Besides, it seems a fact that they doubted that she was the king when they first saw her.
Personally, I’m having a hard time seeing ambiguity in this sentence even without context.