I have one of those adapters that gets used in my various 1903's, both Springfield and Rock Island. .327 Federal cases won't chamber in mine, but I suppose a reamer would solve that problem. The adapter does not seem loose in any of the chambers, but others seem to have had different experiences.
With .32 S&W cases, I use an 0 buckshot, lubed with Lee Liquid Alox, and an appropriate charge of Red Dot. .32 S&W Long cases (Herter's) get a SAECO SWC sized .314 and again an appropriate charge of Red Dot. The buckshot loads are good for empty soft drink cans to 15 yards or so, and the SWC's stretch that to 35 yards on my better days. Primers are Winchester Small Pistol. All the projectiles have made it out the bore so far, but I should mention that a patch wet with Marvel Mystery Oil is run through the bore before each shooting session.
My dies are very old C-H and don't size the cases small enough to use the more common .312 bullets, and don't work with the jacketed bullets I have tried at all. I've tried the buckshot sized to .314 and unsized, about .320 but not really round, and they seem to work the same either way, so I just thumb press them into unsized cases, after priming and charging of course.
That makes for a quick and easy reload.
For me the adapter and .32 loads have been a hoot, sorry I didn't try this years ago.