If the barrel is marked "AI", then it's most likely an Ackley Improved.
It Should have a 40deg shoulder. You should also be able to fireform factory.338/06 ammo (should you find some).
Most likely two dies are the complete set. I don't know what the third die would likely be.
You should be able to reload once-fired .30/06 brass. Running the fired '06 case into the die will expand the neck and set the apex of the case shoulder such that it will chamber in the rifle.
First, load a dummy (empty case, no primer or powder) and minimally seat the bullet, and attempt to chamber it. It shouldn't fit. Progressively seat the bullet deeper by 1/2 turn of the seater knob until the bolt just will easily close.
This is your "fireforming" OAL.
Load the rounds with a starting load of H4895. Alternatively, you could use 10gr of RedDot or Unique. I suggest using either the Speer 200gr HotCore or 200gr Hornady PtSpt as they are the least expensive jacketed bullets in .338.
Or, do as I did making some "useable" .375Ruger brass from .300WinMag range pickups. I loaded and fireformed the cases by using a Lee 255gr cast bullet over 10gr of Unique. The cases look funky with a coke bottle appearance, but load and shoot just fine. I've killed deer with them loaded to full power. If I lose one, no big deal. Not like losing a $2 Factory Hornady case!!!
Use any published .338/06 data as a starting point. Max loads will be 3-8% above standard .338/06 loads.
It'll deliver.300WinMag energy with same trajectory as .30/06. If I hadn't aquired about a dozen other rifles to include two .338MarlinExpress rifles, I'd still be hunting with it.
It's "boringly good". So good and predictable, it's boring. Just like a .338mag without the extra recoil. I haven't hunted with mine in 10yrs or so. Too many other toys.... and our 100-200lb deer or pigs, or 300lb bears, don't need that much killing.