John, welcome to the forum. I think you have answered most of your own questions. Yes, based on the serial number this is a S&W Double Action, Fourth Model in caliber .38 S&W. That is a shorter round than the .38 Special, and has a slightly greater case diameter. It is a little less powerful round than the .38 Special, but it is not a wimpy round by any means. Many British Commonwealth countries relied on revolvers chambering this round during and after WW2.
.38 S&W ammo is hard to find in the United States, but not impossible. Well equipped privately owned gun stores may stock it; a friend of mine got a couple of boxes from the shelves of a store in San Diego a year or so back. If your gun store doesn't stock it (and in an era of chain sporting goods stores, most don't), you could ask them to order it for you.
Alternatively, you could buy it off the internet and have it shipped to you. Different distributors will have it in or out of supply at a given moment, so you have to keep browsing until you find it.
A coincidence: I actually fired a few rounds of .38 S&W through my own .38 breaktop at the range today; mine is the hammerless safety model (double action only); it was shipped in 1907.
That's a very handsome revolver you have. You are fortunate to have it in the family as a generational heirloom.
