But I cannot say plainly
私はアメリカに行くと思っています。
Is that it?
It's not that you can't say it.
This sentence can be considered that the subject either in the main sentence or in the subordinate clause is omitted.
(主語は/が)アメリカに行くと思っています。
私は(副主語は/が)アメリカに行くと思っています。
Both the following sentences are natural conversations.
case 1 example : The subject in the main sentence is omitted.
A:彼はあなた[が/は]どこに行くと思ってるの?(Where does he think you're going?)
B:私はアメリカに行くと思っています。(He thinks that I'm going to America.)
case 2 example : The subject in the subordinate clause is omitted.
A:(あなたは)彼[が/は]どこに行くと思う?(Where do you think he's going?)
B:私はアメリカに行くと思っています。 (I think that he's going to America.)
But can I say, for example,
私はアメリカに行くと思うと緊張になります。 ->Is this sentence structure correct?
(I become nervous when I think of going to America.)
Yes, you can say that way. "(私が)アメリカに行くと思うと" is the subordinate clause in this sentence.
私はアメリカに行くと思うと緊張
します。 is correct though.
彼はアメリカに行くと思っています。 (without making the phrase a subordinate clause)
This sentence also can be considered that the subject either in the main sentence or in the subordinate clause is omitted.
Both the following sentences are natural conversations.
case 1 example : The subject in the main sentence is omitted.
A:(あなたは)彼[が/は]どこに行くと思う? (Where do you think he's going?)
B:彼はアメリカに行くと思っています。(I think that he is going to Ameria.)
case 2 example : The subject in the subordinate clause is omitted.
A:彼はあなた[が/は]どこに行くと思ってるの?(Where does he think you're going?)
B:彼はアメリカに行くと思っています。(He thinks that I'm going to America.)
You need to carefully follow the conversation to figure out which subject is omitted.
I hope you won't get confused.