Hi Jimmy....
What sounds like you have is a "1936" Krieghoff (produced in "1936", as you probably guessed, too <img border="0" title="" alt="[Wink]" src="wink.gif" /> ). I'm going to make a "leap of faith" that all the internals are properly LWaA stamped as well (that small eagle with a "2" under it). However - you don't have what sounds like an HK magazine. It also appears that it's an HK for Military production (based on the proofs/acceptance marks you describe).
However, one small question first, before I go on with a "guesstimate" for you... On the bottom of the barrel, near the "slide" as you describe it - can you tell me what the marks/numbers are?
As reference, here is one of my Kreighoffs I presented to the Forum members a while back - so if you look at the pictures of this one (it's a different production year, but I haven't created a gallery for my "36", "1936", or "1937" HKs yet - but this one should be close enough for comparision) - you should be able to tell me what looks "the same" or may be different on yours (ALSO - if you read the DESCRIPTIONS of those photos, it will tell you what stamps/proofs you are looking at on your HK):
http://gallery.rennlist.com/lugergallery/jdhk38?page=1
Best to you,
John