About the past progressive and present progressive tense.
Which is more natural between the following two sentences?

It would be more natural (US English) to say "I had always wanted..." without the contraction.

"Have always" implies that your deam still hasn't come true. It sounds like you are speaking about something is <ongoing right now>, instead of speaking about something that was <ongoing in the past>.

Specifically the reason past progressive sounds more correct here is because the second clause, "the dream came true" is also something that happened in the past. When the tense doesn't match it can be slightly confusing about when each event started. The other sentence could still work if you changed the last part to say something like "...and though my dream has come true, I still want to go there again"

Your sentence achieves that perfectly by saying, "...finally came true." After the sentence, the reader gets the impression that you are happy.

You could say in your next sentence, "Now that I've gone, I think it's a terrible place," but if you just stop at the first sentence, you seem not to regret it.

I think maybe the more natural way to express regret is to use the past tense. "I used to dream about going to America, until I went two years ago." That sounds a lot more like you don't dream about America anymore.