One never knows the true history of a used revolver. A few inappropriate 'Hollywood' flicks of the wrist to swing the cylinder open or close can warp the crane slightly, making the cylinder turn 'out of round'. This can be observed by emptying the chambers and dry-firing the revolver as you observe - or, better yet, measure with an automotive feeler gauge - the gap between the cylinder's face and the barrel's forcing cone. The allowable limits from S&W are .004" - .010". Whatever it is, it should be constant and not vary from cylinder bore to cylinder bore. Residue build-up on the cylinder face will affect this, too. A trip back to S&W would fix this, of course... maybe for nothing! Call their 800#...
You can shoot the shorter .38 Special's as much as you want in your .357 Magnum chambers. As others stated, you must clean before chambering the Magnums. Buy a .38/.357 'Bore' brush and a slightly larger (and longer) .38/.357 'Chamber' brush, available from Brownell's, etc. The small 'jump' the .38's must make before the step in the .357 Magnum chambers may effect ultimate accuracy, but that shouldn't be a concern in a fixed sight PD revolver, like a 65.
Stainz