Hello Michael5959
You did well, welcome to the forum and also welcome to "The One Liner club" as we commonly call it for our guns with a single line address. As Other's have told you your gun has the single line address that was used up until 1948 when they started stamping all guns after that time span with the new then Four line address. The One Liner Postwar Gun's are far more desirable as their widow of production span was real short being from 1946-1947 time span being just Two short years, before the factory started using the four line address stamp. I paid a Little less for mine than you did being the price of $500.00 at a Local Gun show in my area about Five years ago & mine is a "One Liner Transitional Model", differing slightly from yours.
They call one like mine a Transitional model due to it's Mushroom looking extractor end. These Mushroom style extractor end's were used in the Prewar production revolver's and being that S&W was Not a Company that wasted Part's, they used up the on hand surplus supplies of them in the early Postwar Models, before changing over to the straight wall extractor end's like yours has. It took Far More Machining to use this Prewar style extractor end as they had to Mill out the under side of the barrel recess to allow the larger diameter style head of the gun's cylinder extractor to clear when closing the guns cylinder tightly.

Mine has a Much Lower Four digit serial number than yours being K-366X and it shipped out of the Factory in November of 1947.
If you have not removed your gun's Magna style stock's, the serial number of your gun should be stamped into the inside of the right hand stock. in doing so, I have found a fool proof safe procedure that I always use now & would suggest that you do the same, where you simply loosen the grip attachment screw and push off the Opposite stock panel from the gun's Frame by using the same screwdriver while applying pressure to the screw head, once you have loosened it enough, this Process will work like a Charm, and will save you the problems of leaving dents or damage to your gun's stock's or worse yet the guns bluing finish. Keep in mind that These stock's were hand fitted back then, so they will fit the gun's Frame very, very tightly & I would further suggest that you Never "Pry or Force" The stocks off any gun, Simply use the method I suggested above to Safely remove any S&W stocks.
To answer your question about a refinish on your gun, If it were mine, I would not refinish it, Since the Initials or scratches are not visible with the Magna's in Place, I would simply leave it alone with it's original factory finish still In Place as any refinishing work will drop the value of your K-22 to Half of the market price. But,
"it is your gun so if you desire to have it refinished it is totally up to You", and for doing so I can not think of any Better refinisher's out there than
"Ford's Custom Gun Refinishing". They are located in Crystal River, Florida and do excellent work, see their web site link below as I have had several done by them in the past and they have always by Far Exceeded my expectations. I am Not a Spoke Person nor am I Paid to advertise for Ford's Custom Gun refinishing, I feel compelled to give Credit where Credit is do,and Ford's certainly deserve a Pat on the back for their attention to detail and fine Craftsmanship.
Shown below is my 1948 Four Line address K-22 that I sent to them about Five years ago. I bought it dirt cheap as it had very Deep Rust Pitting and careless handling wear on it, which can be seen in the Before & after pictures below of it. This Gun shot Teriffic and I felt obligated to giving it the finish that it deserved, so I sent it down to them along with a New Old stock Factory McGivern Gold Bead front sight blade and they installed it for me as well before rebluing the gun. There was a slight problem though that I missed when I sent the sight along with it, as it was made for a later wide rib K-22 from around 1951 or so time span when S&W changed the barrel rib's to be a Tad Bit wider so it has a few ribs too many on each side of it making it wider than my Gun's earlier made Thin barrel profile .
Horace Ford, the shop's owner & master gun smith took my sight blade and Milled off the extra width so it fit my barrel perfectly, these People really know their Craft and go the extra distance other gun refinishers will not bother to go. When I sent it to them, I asked Ford's to place Their Master Bluing finish on it which is very High Polished deep Black style of bluing even though a gun from 1948 would not have come like this,
"I wanted the High Polish bluing on it to suite me", Not caring what was correct for it.
In Having Ford's refinish your Gun rest assured it will come back looking great. Unlike other refinisher's out there, Ford's take the entire gun apart removing Pins and cylinder stop's so that they will not look Mauled or flattened, like other refinsihers make the mistake of doing in the refinishing process. They also hand polish them where the factory stamp's are as to not weaken the factory stamping's, by Machine buffing a common mistake made by other refinishers out there and they never round over the frame parts or dish out the guns side plate screw locations which we commonly see from other refinishers out there that are in a Hurry, not paying close attention to the fine details or Pleasing eye appeal of their refinishing effort's. They also do all of the finish work for Desert Eagle and Magnum research guns, and as far as I know are one of the only Refinisher's out there to offer a Year warranty on any Gun's that they refinish, that tells you how well they do their work, as other's in the gun refinishing business do not offer any finish warranty.
One Good thing is that You can ship your gun to them directly without getting skinned back on shipping cost's Imposed by UPS or Fed-X by simply sending it through the U.S.P.S. Legally as they are a Gun rebuilding & Repair facility such as the factory is and they in turn will ship your gun directly back to you without having to go through a Licensed FFL dealer, that seem to always Gouge your wallet for a Transfer fee of an average of $35.00 and a Ten dollar back ground check done that one has all ready Paid for when buying a gun which cut's the over head cost's of having your gun refinished at Least $45.00, Plus the additional savings of not having to ship it over night imposed by UPS or Fed-X which on average is now about $50.00-$70.00 saving you an average of 115.00.....
Now, you will have to get busy and locate the Remainder of the "Masterpiece Trio" by adding the Two that you are Now lacking being the Hardest one to locate of the Three revolver's the Elusive K-32 Masterpiece chambered in .32 S&W Long Cartridge & the K-38 Masterpiece series chambered in .38 Special caliber , as I finally was able to Complete the Five screw variation Pre-Model stamped Masterpiece Trio of these revolver's a couple of year's ago myself, and am Glad the Long Tedious hunt is finally over for me, as well as the cost Involved in it.
Here is my One Liner Transitional K-22 Shown with it's original Box, Paperwork, & Tools shown with it's original serial numbered Magna's as well as with an early style Diamond center Target stocks made from the exotic Gancalo-Alves wood, that it is currently wearing now, as they fit my hand better and I feel they look more at home than it's Original Magna style stocks that it came with. Your Gun would have come in a Gold Postwar style K-22 Box like mine, with a Knurled Black oxide finish sight adjuster "AKA Screwdriver", or possibly the Nickel finish sight adjuster as S&W Placed what ever they had on hand back then so these sight adjusters can not be conclusive and it would have had an aluminum or Blued Steel cleaning rod, a Bronze bristle cleaning bore brush, a cotton cleaning bore swab, a Postwar style Ten ring tip's brochure showing the Officer standing in a shooting stance with slacks on, & a warranty card as well as dry fire Practice target like mine shown. I hope this helps answer some of your concern's, enjoy that fine early postwar One Liner K-22 and Please let us know how it shoots and groups. Best Regards, Hammerdown
Ford's Custom Gun Refinishing - Ford's Desert Eagle Sights
The Before and After Pictures of my 1948 K-22 that Ford's Custom Gun refinishing Placed their Master Bluing finish on
Before..... shown Badly Pitted and abused as far as it's finish went.
After They were done with it, shown with it's new Master Bluing finish