I own the 651 (the shrouded-hammer version of the 650) and it IS a little bit heavy. Not a big deal, though. I carry it daily (usually in an ankle holster.)
.357s DO kick more than .38s, but it's not that big a deal. Of course, I'm not a particularly small fellow, but I don't think the recoil will be unmanageable for most folks.
I carry Speer Gold Dot JHP cartridges in it for SD. You might try the Short Barrel Gold Dots, too. (My range was out of 'em last time I went.) They're designed for short-barrel guns like the 650.