Can my guard unit legally delay my paperwork? What recourse do I have?

If it's a 368 that means it is a Conditional Release and you are requesting to be released for some purpose, not demanding to be released under current policy. A 368 is never required to be approved, if it was required to be approved it would not require approval.

Obviously no person here can consult you on your ADSO since we haven't seen your records, but generally a DD214 does not state the ADSO of an officer, it only lists periods of active duty greater than 90 days

First, the bad news. Your unit is under no obligation or time schedule to sign you DD368, which should require the signature of the first General in your Chain of Command unless otherwise designated. Some States require the signature of the TAG.
If you are a Guard/Reserve Commissioned officer, your initial Obligation is 6 years + 2. If you are prior service, you enlisted time counts toward your total MSO.
I would talk to my commander about the chance of hand carrying your DD368 to the General that's signature is required. You still may get told no, but you will at least know it up front.

You are unlikely to get anything done by Sep 1, unfortunately. If you give one level of your chain of command adequate time to take care of a request and they don't, you simply go to the next level. Next time you're in this position, don't be afraid to walk up the chain if you are getting stonewalled.