Have you fired the 637?
No doubt in my mind that a J frame is a timeless classic. I carried a M36 (not an airweight) for many years. Reasonable accuracy and not unpleasant to shoot with a T-grip filler. For my purposes the only shortcomings were the hammer spur which will reach out and snag on anything within three feet. And in South Texas it is always a battle to keep a blue gun corrosion free (but stainless was just coming in when I bought this).
A couple of years ago my DW surprised me with a 638 I had admired at a gun show. A fine weapon, no hammer spur and stainless - just two shortcomings in my book.
1) it is decidedly unpleasant to shoot. I'm not a total pansy, but it is not fun. Not recoil so much as sting. Think snapping a big rubber band across your palm.
2) no one expects a J frame to be a tack driver, but this thing looked one way and shot another. I thought I must be pulling because of the sting (as well as the short sight radius). However, after a I put on a set of CT grips (which are normally zeroed out of the box) and adjusted them (a bunch) it seemed to shoot as well as my old M36 so I don't think it was me totally.
The M36 is still in the safe and the 638 or a CS9 are my EDC, but I don't look forward to taking the 638 to the range.
With all that said, I'd jump on the 637, too. Just be aware if you haven't fired these airweights.