The standard "factory" ammo in .44 S&W Spl. is built to the original spec--IOW, no harm to your 629. Even "defensive" issue is not a problem; this round runs well in lower pressure ranges compared to .357 Magnum rounds.
However, as you may have discovered, it is hard to locate .44 Spl factory ammo--and when you do, you will not want to pay the price. So, a lot of us reload...
And, if you have not started reloading, this round would be an ideal one to learn on. Big bullets, big cases, good components readily available--the only caveat is to watch for double charges in that big case.
A perfectly safe load for your 629 would be "Skeeter's load"--and if you are interested in reloading, look it up--and come on over the the Reloading forum here. Meanwhile, don't worry about any .44 S&W Spl ammo in your 629.
Jim H.