Sorry it took so long to reply but I have been a busy man lately. The serial numbers on my model 37 are 307xxx.
According to the SCSW 3rd ed., your Model 37, .38 Chiefs Special Airweight's serial number would place it between 1962 and 1969. That S/N is not very precisely detailed in the book. They began stamping the model number 37 on it in 1957, and it did not go to 37-1 until 1988, even though there were several "engineering and production changes." The changes listed may help to narrow down the date of production for you.
1962: Internal hammer alteration.
1966: Thumbpiece change.
1968: Delete diamond grip.
1969: Begin J serial prefix.
Edited to add- The S/N listed for 1962 is 295000, and for 1969 it is 786544. Your 37 is probably very close to 1962-63.