Shot loads are typically not very effective except at rather close range - closer than you'll typically be able to get to small game.
Father-in-law uses 'em on squirrels that are raiding his bird feeders. He kills one now and then (with multiple shots) - but typically just peppers 'em enough to run 'em off (temporarily).
If you "pattern" a shot load out of your 44 on something like a picnic plate, I suspect you'll find they get pretty useless beyond 4 to 5 yards.