AFAIK the velocities will be very similar in shot guns, wether it be a 20, 16, 12, or 10 gauge ect.
Its the PAYLOAD that they can shoot.
the advantage of the 10 gauge over the 12, is not velocity just a larger/denser pattern of shot. so i suppose it could give you greater range since the pattern will be more dense futher out.
I beleive the saying 'its not the arrow, its the indian' is appropriate.
If you can't kill it with a 3" load out of a 12 gauge, or at least knock it clean out of the sky, i dont think a 10 gauge is going to help.
I'd suggest changing chokes, waiting for the game to come a bit closer, and changing technique is what should be done, not just blaming it on the gun.
Not to say there isnt an advatage to 10 gauge in situations but thats just how i see it.
Kick wise, just look at the payload weight vs the gun weight and there you go...since like i said before the velocities should be similar in all.