I thought blah blah blah, and I was right. Is the comma correct?

The comma is enough as the second clause "I was right" is independent.

Commas are used before conjunctions (but, and, yet, or, so, etc.) when the two clauses they are coordinating can stand as independent sentences.

The semicolon is used when the first clause contains commas.

I knew going in that the orange, marzipan and chilli flavor component of the cake was going to be lacking; and I was right!

They are both right. As is

I knew going in that the orange flavor component of the cake was going to be lacking. And I was right.

Your examples are both compound sentences. There are two independent clauses, joined by a conjunction (and) and separated by punctuation (either the comma or the semicolon). The comma is a softer break, the semicolon, a more significant one.

My example uses two separate sentences. This is an even more significant break and is somewhat more dramatic. A few purist would argue that sentences should not begin with a conjunction, but that viewpoint is probably not very ascendant today. Or do you disagree?

[Perhaps I should have begun this entire discussion with They are both right, as is ...? Or maybe They are both right; as is ...?]

A comma seems fine. A quick look at google books shows that commas are common here, though sometimes there’s no punctuation at all.

I found one case of parentheses, but not a single semicolon.

The Economist’s advice on semicolons is to use them:

to mark a pause longer than a comma and shorter than a full stop. Don't overdo them.

So, it’s seems that a comma is fine in your example.