You can use a comma or a dash to connect these pairs of sentences, but writing them separately is not incorrect. It is looked upon by some as informal.
He started a sentence with a coordinating conjunction. And that was the end of him.
He started a sentence with a coordinating conjunction. But his wife didn’t leave him.
He started a sentence with a coordinating conjunction. Or perhaps he only dreamed that he did, because the kale was spoiled.
Does OK mean grammatical or stylistically acceptable? This statement from an Oxford Dictionaries blog addresses the question:
[T]his is a stylistic preference rather than a grammatical “rule.” If your teachers or your organization are inflexible about this issue, then you should respect their opinion, but ultimately, it’s just a point of view and you’re not being ungrammatical. If you want to defend your position, you can say that it’s particularly useful to start a sentence with these conjunctions if you’re aiming to create a dramatic or forceful effect. (“Can You”)
“Can You Start a Sentence with a Conjunction?” Oxford Dictionaries Blog, Oxford UP, 2019, blog.oxforddictionaries.com/2012/01/05/can-i-start-a-sentence-with-a-conjunction/.