The Go No-Go and Field Gauges
When building a rifle,the Go must Go. The function and use of the No-Go gauge is to tell the gunsmith "The Go went,the No Go did not,your work is done"
The No-Go is NOT for pass/failing a used gun. The field gauge tells you its time to take it out of service.
"The bolt closed with a little effort" When using a plug gauge to check a diameter its held between the thumb and forefinger. "A little effort" is not used.
Your headspace gauged "good" . Your rifle is fine (probably)
Did that ammo come as a package deal with the gun?
In my limited experience with 30-06 factory loads they tend to size cases to the minimum. Likely for the semi-auto Riminton Deer Guns that sell small based dies.
Tolerances stack. You do have a chamber that leans toward "Max" so cartridges that lean toward "Min" stack a lot of stretch to the case.
I've had a similar problem with a 7mm Rem Mag. Rifle gauged good but the tolerances on belted cartridge headspace tend to stretch brass excessively...and cause stretch rings.
I learned to abandon the belt for headspacing ,fireform the brass and use the shoulder.
Advanced handloading techniques give us the option to do these workarounds.
But thats a lot to grasp for a self proclaimed newbie.
It can be helpful to know where you are before you route where you are going.
At risk of stirring up the crowd who hates the Wilson Bushing Guage, I'd buy one. Many folks hate what they do not understand.
It has a step ground into it that will show which side of the tolerance any ammo you have might be.
More use later, with the case in the bushing gauge you can use your calipers to measure over the case and bushing. That number from a fired case is useful.
When you reload,you can minimize the shoulder setback to reduce additional casestretch.
In a bolt gun,.002 is a good head clearance to shoot for.
The use of a feeler gauge between the shellholder and sizing die is one method for a repeatable die setup.
I don't like the idea of using 270 or 280 brass necked down a bit to create a trace of false shoulder that would hold you brass back against the bolt face...
it would work but getting in the habit of shooting ammo stamped "270" in a 25-06 invites a lesson in Murphy's Law. Bad idea.
But if it was me,I COULD buy virgin 25-06 brass, neck it up to .270 or 7mm,
then partially neck it down to get the same effect.
Or load some reduced loads with the bullets seated long enough to crush into the lands.
When we fireform virgin brass to fit the chamber, then have the means to control our resizing to minimum head clearance, we can handload around your case stretch problem.
The "Cream of Wheat" method of fireforming was mentioned. I have used it with good success to form wildcat cases. It works. Important!! You DO NOT use a bullet!! A small (approx 10% of a normal rifle powder maybe 5 gr?) charge of Bullseye is loaded in the case. You will have to experiment with how much Bullseye is needed.
Then the case is filled with Cream of Wheat. A pinch of cotton or toilet paper can be used for a "cork" Then set them in a loading block. Go some place you can shoot them,single load them without spilling and shoot them muzzle up.
It works but its pretty involved. Not my first choice!!
When 30-40 Krag and 303 British were readily available I was blowing them out to straight wall for Black Powder Cartridge 40-60 Maynard cases.
I made and chambered a chamber insert for a break action single shot shotgun as a fireforming tool. It worked great!! Then the .405 Win case was offered by Hornady. Hello,40-70 Sharps straight!
I don't recommend you pursue the Cream of Wheat method for your 25-06.