The accuracy, of .38 ammo, is ever so slightly worse from a .357 firearm than it would be from a .38 only firearm. This is because of the extra fraction of an inch that the bullet must travel before meeting the sweet embrace of the forcing cone.
The real issue is the cleaning procedure for the chambers.
The Blackhawk is an excellent firearm for shooting both .38 and .357, since you can easily remove the chamber. You need a stainless-steel .357 chamber brush. Regardless of whether you intend to fire .357 ammo, every few 100 rounds of .38, you should dip the chamber brush in Hoppe's #9, or equivelent, and force it through each of the six chambers. Some people actually use the chamber brush with a power drill. This is OK for a revolver chamber, but never, ever, for a barrel, or the chamber section of a pistol.
Lead ammo is perfectly fine, but I'd recommend shooting a few 100 rounds of FMJ through the barrel first, to smooth it out. For .38, look for 100gr. lead wadcutter for mild loads. If you're not "recoil sensative", 148gr. TMJ wadcutters are reasonable target ammo.
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