He told me that he did not go to church every week.
He told me, he did not go to church every week.
One, both, or neither?
I don't think you need to use it sparingly. To me, it reinforces you are using reported speech, and not a direct quote, but it's completely optional. Its use is a matter of style and writer preference.
I'm not a teacher, but I write for a living. Please don't ask me about 2nd conditionals, but I'm a safe bet for what reads well in (American) English.
There may be a tendency in spoken English to eliminate "that" whenever possible, but in written English it can make things clearer. As Barb_D says, it is very much a question of style.