As John Beardsays, pinned stocks are known to split. Check yours for any splits at the tang (should be about .05" separation between tang & wood) and inside the stock where the recoil lug seats. If the stock is undamaged, consider replacing it. The CMP
is selling original WWII mfg walnut stocks.
If there are spits present, you can repair the stock with epoxy or acraglss. If you need to do that, let us know. A properly repaired stock will be strong enough so that it does not need to be replaced & will make a good shooter. The stock has nice color & figure, by the way.
When the rifle is out of the stock, check the bottom of the receiver ring for any sign of welding. 1903s were converted by the Army to Drill Rifles by welding the receiver to the barrel; driving a plug into the chamber; welding the cut off switch closed & welding up the bolt. Your cut off looks fine but double check to see what's underneath. A weld may show bubbling; discoloration & at the barrel/receiver joint, bright metal like stainless steel as shown in the pics. Some people convert DRs back into firing shape & don't bother to tell buyers.
Attachment 15274Attachment 15275
I'd also check the headspace, especially if you replace the bolt. The Army never withdrew the straight handle bolts (most of which received the "Single Heat Treatment" and may be somewhat brittle. A few received the "Double Heat Treatment" and are fine. At Camp Perry, shooters with SHT bolts were advised to bring a spare bolt for their rifle that was already headspaced. That way, if the lugs failed (safety lug held), the shooter could remove the failed bolt, insert another & keep going. John has a Bolt Heat Lot chart at http://www.vishooter.net/m1903.html Match the marking on your bolt (on the root of the handle or safety lug) & you can date your bolt. If the bolt has a straight handle, it may be a desirable collectors piece & you can buy and headspace an A3 bolt & then sell the straight handle bolt.
Also, avoid the fish-based lubes; the cat may get a whisker stuck in the action & they get upset if they loose one.