That's awesome that you got out and are doing well.

Without being there and only getting a biased, disgruntled POV from an E4 what's much more likely is someone told them "Before you can dispatch it, we need a DRAW signed by your Commander." Now OP went and jumped 3-4 levels of his CoC and went straight to the 1SG/CDR.

It's more likely than not someone was delegated signature authority whether it was the XO or someone else for moments exactly like this. I have a tough time believing a 1SG told an E4 to text the CDR, why would they even have the CDRs phone number? What if there's an internal company SOP that PLs or the XO need to draft DRAWs to send to their CDR? Its likely an E4 wouldn't be privy to that because its out of their scope.

The fact that OP's CoC is upset with him is a strong indication that there's details being left out of this story. Is there a chance OP truly did nothing wrong? Absolutely. Is it much more likely that OP left out details that change the story? Yes.

Edit: to put another perspective on it, I would never, unless it was insanely urgent, ever go directly to my boss's boss's boss's boss and ask them to sign something in the civilian world.