You need a varmint shotgun. I'm doing research on one right now--I need a lightweight one. Went into the gun shop and asked about one (I joined the NRA to spite the Clintons in the nineties, not because I know a thing about guns) and got asked a bunch of questions, so now I have to find the answers.
how old are you, how tall are you, how much do you weigh, and are your shoulders in good condition?
I probably do but I love my dogs too much to deprive them of the fun of running them off.