ok,
lets be the devils advocate...first,, SARCO is known for selling used parts as new, as well as import reproductions as GI...amoung other things, likely thats the case....all new replacement A3 barrels made after 9-43 were short chambered.
another issue you may have is a stretched receiver,, id need more info on that, to be more correct.. pictures, maker, serial ect.
and finally...
this whole thing of just turn the shoulder back,,,brings that headspace right up...its easy.... NOT!!
anyone who says that has never done one.
heres how that happens.
you turn the should back just enought to get one complete turn from the barrel, remember the A3 has a square thread, so your taking a lot of that shoulder.
but you forgot the big picture....ok, now you taken the shoulder back,, now my bolt wont go in....sure it wont.... you have to turn the chamber face in the same amount that the shoulder was done, then recut the bolt notch on the right side of the chamber face...remember, the A3 has a coned breach...and you have to use a special cutter to do that job..
then walk over to a mill and recut that extractor notch... no mill? will get out a hand file, tape the edges accross the notch so you dont bugger it up, and file the notch,.. might get lucky and get that done in a hour,
then... re cut the chamber, only now, you may have a step as the old chamber was likely over sized a bit...
so,,,, now, figure an hours lathe time and 2 set ups. most charge 75.00 an hour. plus a 45.00 set up.
now add an hour mill time or hours labor....75.00 in most cases...
then reinstall the barrel... thats another charge, and finsih ream it again..another charge...
why all these charges you ask? this wasnt the gunsmiths fault, he did what he was suppposed to do, and from what you said,, did it right.. id bet if you used a GI set of headspace gages that it would gage right up, and likely be a little short.
if your barrel pased field reject then it is safe to use, your bolt may be the issue as well, many bolts are worn from manual of arms training...
one question...what maker is the receiver???Remington Smith Corona or other.