In these cases, ones without だ look more or less clumsy to me, because the part before と is essentially a sentence with a verb and a corresponding subject.
(quoted sentence with plain form ending) と (Your verb. i.e. 言う、思う、聞く、etc)
So really what comes before the end of the と particle in this case is the plain form. I think that several examples will help you out.
彼女は綺麗です。(She is pretty)
綺麗 is a な adjective. Using it at the end of a sentence, we get です, or in plain form だ instead of な。
彼女は綺麗だと思います。(I think that she is pretty.)
Taking the first example sentence, and putting it into plain form, we can add と思う。 Without any complications.
あのビルは大きいです。(That building is big.)
大きい is, of course, an い adjective. In the sentence I ended it with です (to be polite), but です in this case is unnecessary. I could say あのビルは大きい and still have it mean the same thing. Plain form with い adjectives actually end with the adjective, which in this case is 大きい。
あのビルは大きいと思います。 (I think that building is big.)
I'll give you a few more examples without commentary, to hopefully solidify the point.
あの人は大学生です。(That person is a college student)
あの人は大学生だと思います。(I think that person is a college student.)
彼は十キロ走っています。(He is running 10 kilometers.)
彼は十キロ走ってると思います。(I think that he is running 10 kilometers.)
Additionally:
A verb clause can be phrased by adding 「と」 to the end of the clause. For clauses that end in an plain noun or na-adjective, we must add 「だ」. A direct quote would use the Japanese version of double-quotes: 「」 but you can also paraphrase.
Thanks for taking the time to thorouhly look into this.
The case I used in my post seems to be an edge-case. Since I don't yet know what sounds natural in this situation, I guess I'll just have to encounter it more in my immersion.
Since I wasn't aware of this grammar point until now (didn't study grammar on its own much) I appreciate you making me aware of it.
Have a good one :)
PS: obligatory "English is not my native language"
Sure. People will end up using language in written and spoken forms differently. I just wanted to properly address your point when you said you've never heard of this rule, and could not find anything regarding it.