Congratulations! Great stock markings and I’m sure you’ll enjoy it. I really like the 03-A3. A great piece of WWII history and a very good practical shooter with the rear peep sight. The Model 1903 is an elegant rifle with a sight system designed for highly accurate and precise target shooting. The 03-A3 design changes to reduce costs and speed production also encompassed that change in philosophy toward practical combat shooting in WWII.
Finding a correct one is very rare, though a “corrected” one is more common. Definitely need to disassemble it and examine and/or photograph all of the parts and their finish. There used to be an online resource for bolt handle lot numbers, but I haven’t looked in quite a while.
I have the three manufacturers of Model 1903s (the SA is a Mark 1, no Pedersen device, of course) and the two of 03-A3s, as well as an 03-A4. Called it good when I checked the last one off the list. Like you, my first was the Remington 03-A3 and was the closest to being original and correct, but it wasn’t.
I worked as a civilian GS-12 term at Tyndall AFB from September 1993 to 1997, my first postdoctoral fellowship. I was over on the non-flight-line side of 98 at the lab on 139 Barnes Dr. I wasn’t into guns then and had zero extra income anyway

so didn’t know anything about the local gun shows or shops.