Shoktronic re Model 36
The number on the bottom of the grip 237XXX is the serial number. The other numbers are assembly/parts numbers. The model was originally called the Chiefs Special and was the first to use the new "J" frame size. S/N 1 was completed on October 24, 1950. When S&W began using model numbers in 1957 the Chiefs Special was assigned model number 36.
I don't have exact year by year dates for the model 36, but can narrow it down a bit. The first of the Chiefs Specials to get the model 36 stamping was s/n 125000 in 1957. S&W made a change in how the mainspring and hammer connect on Sept. 12, 1962 at approximately s/n 295000.
A reasonable estimate for year of production would be 1960. The grips you describe would be correct for the time frame. These compact J-frame revolvers were tremendously popular when first introduced and remain so to this day.
Assuming it is in sound condition it can safely be used with any standard pressure .38 Special ammunition.