Spanky,
I've shot some factory 44 special jacketed loads in my handi. How well they will work depends on your rifle, but I was generally unimpressed. Be aware that because of the handi's slow rate of twist (1:38"), slower moving bullets may not have enough RPM's to stabilize. I haven't shot any factory lead bullets, but I handload cast lead bullets using Unique. Generally, I get adequate to good accuracy if I load 200-210 grain bullets to a velocity of 1200 fps or more, and 240 grain bullets to a velocity of 1350 fps or more. Seems like the shorter bullets don't need as much velocity to stabilize, or maybe its just my imagination, but these benchmark velocities generally seem to work for me. With lead bullets, bullet diameter can also be a factor. Although "new", my 44 barrel is a little overbore (.431" diameter). It will shoot .430" jacketed magnums just fine, but definitely prefers .431" or .432" diameter cast bullets. I'm not much of an expert, but I hope this helps.
Duane