Aloha Kelvin...
"Free floated" systems on M14's and AR's are different. As Warbird said, the match free floated M14 type rifle would still have contact at the front ferrule and the front band.
Whereas the AR systems, the barrel and gas system is entirely independent of the handguard.
I am sure if you squeeze the handguard (gently) and the stock you will notice some compression there... that draw pressure is there for a purpose. It allows the barrel and gas system to return to its original state quickly and regularly. So your question on "...does the stock push on the barrel?" No, but the stock does "pull" on the front band (attached to the barrel assembly) to create this pressure or draw.
The thicker SM/NM SA stocks should allow you to sling up tight and not affect zero. Check your front sling swivel (it should be welded shut). It has been a problem with folks who sling up tight to open up the front sling swivel and thus, welding it shut (from coming apart) is normal on a NM rifle.
I hope this helps some...
Aloha,
Tom O.