While I understand the point you were trying to make about the original example sentence and its incorrect double use of “Don’t”, this correction you presented is also incorrect. There are places you can go (such as here ) to find the complicated explanation about the simple fact that we do not match the tense of the main verb to the tense of the auxiliary (helping) verb “Did.” Rather, we use the simple base form of the infinitive (the verb in its “to” form, e.g. to go).
Using “to go” as an example, the basic past tense would be “I went to the store.” However, when adding the auxiliary verb ‘did’, we would NOT say “I did went to the store.” Instead, it is “I did go to the store,” or “She did go to the store.”
Therefore, in your example, it should be, “Didn’t you say you don’t like me anymore?”
You might notice I removed the “that” from the original sentence, as it is not necessary. In fact, since we are in this thread, one of my personal pet peeves is the overuse of “that” in writing/subs.