Missouri bullets is the way to go.
Missouri Bullet Company they are open, have stock, good prices, and proper bullets for original 38 S&W, 145 grain RN.
You can shoot .357s if you want, but unless you have the right bullet, you will not have anything to compare accuracy with. Lots of people shoot 357 and I have compared accuracy in all my 38 S&Ws. Not one of them shot the small bullets as well as the .361" bullet.
I went through that process many years ago and first important issue is land diameter in a 38 S&W. While barrels from 38 SA spur triggers, to 38 DAs, to 38 Perfected, to 38 HEs vary slightly, the average land dimension was .355 to .356". That is just not enough to grab a bullet. The cylinder throat diameter in 38 S&Ws is .361 - .362", so the .357 bullet rattled through the throat and can end up with a shaved bullet and leading when it hits the barrel slightly mis-aligned. The case for a 38 Special lays in the bottom of the cylinder, also affecting accuracy when fired. Lastly, there is a loss of velocity with solid base bullets, since gases can escape around the bullet through the grooves. None of those issues help.
Lots of people just want to go out and shoot the gun, so if it has been re-chambered for 38 Special, no harm done. If one wants to recreate the original ammunition for this caliber, use the right components and accuracy will be improved. One caveat is that you can use cut down 38 Special cases or in the case of lengthened chambers, full length. After shooting the first time, the brass will be re-sized to seal in the chamber of the 38 S&W. Get a $25 set of 38S&W dies and you are set to go. Problem with using 38 Special dies is that they will resize the brass down to 38 Special dimensions if the case even fits into the mouth of some die sets? Also, the lead will be slightly downsized when forcing it into the .357 case and resized even more when crimping the bullet and finishing the case, if the 38 Special dies even can be set that low.
Why take the time and effort to produce something that is poor quality, results in potentially poor accuracy, and not as originally intended and designed?