I just realized today that for some reason I'm pretty uncomfortable with things like そう思う because the Japanese そう has a meaning akin to saying isn't that so in English. But when you say そう思う, そう is being treated as what's being thought, right? So is it more appropriate to say そうと思う？Because it is the subject in this case, right?
The そう is 副詞/an adverb (meaning #1 in そう【然う】) and the subject for the verb 思う can be 私, あなた etc. (can be omitted), depending on the context. Like, わたしは*そう*(≒そのように/そんなふうに)思います。I think so / that way.
I think you are misunderstanding the grammatical use of そう as in そう思う、そう（だ）and other "fundamentals" such as the use of と（as in と思います, という etc)：
そう＆こう are used to refer to the previous sentence and are used with a verb like an adverb.
Here you are stating what you think is correct but you are not certain.
そうと思う might be understood but I don't think it works for reasons explained in the accepted answer to the following question: