I'm not sure I understand your circumstances correctly, but lets walk through it.
Are you looking at documents that LZ is showing you? Confirm that LZ is showing you the correct documents from the correct file. There should be a reference to the corporation id number somewhere on the document. This may be a computer error on their end.
It wouldn't be surprising if this was a mistake on the SOS' part. I would recommend you call them again. You may have to again re-send the documents to the SOS, so get the representative's name and ask if you can work with them directly (and be sure to send it out ASAP). You could probably ask LZ to do this for you.
The fact that there is a 6 week delay to rectify the mistake is odd. I've raised similar issues with the SOS before and I routinely get rectification within 24 hours. Confirm who told you it would take 6 weeks to rectify the problem.
Also try calling back and getting another representative as you tend to get varying personalities with some more helpful than others.
You're not the first person to have some problems with the SOS, just understand the circumstances and keep trying.
You feel like suing them incompetents.
If that's how you spoke to them on phone, that's why you are waiting.
They are very busy, small typographical errors to happen. It now takes about that time for a mailed copy of a new LLC to be filed. Keep checking in and the matter will be rectified.
The above is general legal and business analysis. It is not "legal advise" but analysis, and different lawyers may analyse this matter differently, especially if there are additional facts not reflected in the question. I am not your attorney until retained by a written retainer agreement signed by both of us. I am only licensed in California. See also avvo.com terms and conditions item 9, incorporated as if it was reprinted here.