Yes DA and your BN and BDE should be tracking NCOERs. Back in the day I had to turn them in the 'first day possible'= ie 30days before the rating period had even ended.
You need to fix 1SG and your Commander and tell them you're not doing someone else's job. You should fix the orderly room/ S-1 so they are tracking when they're due and telling whoever the supervisor of the raters are that they need to get them done.
The fact that someone who is not the rater is writing someone else's NCOER means they're not getting an accurate NCOER.
Have some NCOPD to teach the NCOs in your unit how to write NCOERs, when they should be written, types, etc...