Are there CA state withholdings but no state ID?  If no state withholdings just use the Fed EIN in that box and Efile it.

If there are CA state withholdings and NO CA state ID number is shown, the CA refund will be delayed, FTB wont be able to match up the withholdings on the W2 with what's been reported to them by EDD.  The employer has to have an ID number in order to pay in the withholdings to EDD...Id be calling the employer to get that ID number.

Yes, you can choose to only e-file Federal, and not e-file the State.

As you say, you uncheck the state when you e-file (or better yet, indicate in the information worksheet it is not going to be e-filed).  Are you having problems with that?  If so, please give details about whatever the problem is.

I am attempting to do this on a 2020 return, but when I uncheck the state in the efile wizard, fed also unchecks - all or nothing it seems like.  Have done this many times in past years.  Any suggestions?