It's not wrong, but there's no reason to use "did" after "but I". There's also no need to say "the money". Here's my version: Not only did I get charged twice, but I also received the package six days after the date I was (or "had been") told (or "promised").

Not only was I charged twice but I also didn't receive the package until 6 days after the promised delivery date

Not only was I charged twice, but I also received the package 6 days late. Or you could also say: Not only was I charged twice, but on top of that, I received the package 6 days late. (the second sentence is the best choice)