Subject: Re: Trump Signs Order To Eliminate DOE
...consistent with law...

From what I know, that isn't what is happening. As you say, they have to abide by the APA. Perhaps the latest EO directs them to do that? If so, then it likely is legal (if stupid). I haven't seen the text of the EO, but I suspect it's more "take a chainsaw too it, and let them try to stop you" than "follow the APA to systematically shut down the agency". That's his style.