Whats The Point?...

You should talk to Allsup or a lawyer, but technically, you don't need a diagnosis because they don't base your ability to work on a diagnosis. You have to prove that you can't work full time at any (sedentary) job, that you have experience and training in or could be retrained to do. I think there is a category for unknown or undiagnosed or something.

A functional capacity evaluation, possibly a neuropsych test, especially if you have lot of brain fog or mental health issues, and a record of continuing disability over time are good medical evidence to have in your file to show that you can't work.

Also, applying for unemployment while applying for disability is like putting on foot on the gas pedal and one foot on the break, in that you're saying that you both can and can't work full time.

I just googled it and saw varying advice, but the best answer seemed to be to talk to a lawyer before doing it. You are lucky that you have California short term disability, which you can get quickly. It lasts a year.

Also if you had disability through work and they terminated you instead of asking you to go on disability, maybe something could be done there too.