Some thoughts, and lots of questions.
My feeling is that you have a problem, but you have not told me enough that I can be sure. Can I suggest that before you do anything else, check the calibration of your micrometer or vernier caliper against a gauge block or a factory jacketed bullet. The cheap ones I have, have issues, generally measuring about .002" too large.
Also, Can I suggest you load several dummy cartridges with as-cast diameter bullets and see if they chamber. Can I suggest you also see if the as-cast bullets fit snugly in your revolver cylinder throats. Throat fit should be really snug, and Lead bullets should be .001" or .002" larger than barrel groove diameter.
My guess is that there is a lot of Antimony in your scrap, which makes bullets cast bigger. Antimony does not harden Lead much, except when there is Tin also present. Lower casting temperature may also give you a smaller bullet. Using a graphite mold release will give you a smaller bullet, by something in the neighborhood of .001":
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The more you size bullets, the worse they generally shoot. I like to load my cast bullets about .359" in the .38 Special/.357 Mag, determined by what the throats are on my revolver cylinders and what will chamber. You may be able to get away with .360". If you load too large a bullet in a case, the bullet will not chamber.
From what you wrote, I am guessing you are using Lee Tumble Lube and the Lee sizer. Using the Lee TL system, a too-large bullet will have its lube grooves seem to go away with a lot of sizing through smearing. If you are using the Lee system and wish to stick with it, I would get a larger ID sizer. My recollection is that Lee does not make a .359" or .360", but will make a custom one for you. Be aware that running a too-large bullet through a .359" die may not give you a .359" die (eg I have .366" die that, when I put .368" bullets through it, gives me .368" bullets).
You might also simply Lube, but not size, load normally, then use a Lee Factory Crimp Die to bring the cartridge back down to a diameter acceptable to your chamber.
Lubing in a RCBS/Lyman/Star with a lube such as 50/50 NRA formula through a die at as-cast diameter, then sizing down will prevent grooves going-away.