Take another look at the original sentence:
You wanted to fix it this way:
A comma splice—a major sentence error—occurs when you have two main clauses connected with just a comma. Mrs. Britton was grading papers at her desk is one main clause. Her students were slaving over their in-class compositions is a second main clause. The conjunctive adverb moreover doesn't change the fact that you still have two main clauses connected with a comma alone.
You might want to consult the rules for fixing comma splices and fused sentences.