There is nothing wrong with a 22 for coyotes. I know for a fact it kills them. That being said I agree with 20ga buckshot. Go look at the mossberg maverick. They have a plastic stock and should be lighter. I think they come in 20ga. They should be around $200. Big 5 has them for sale often (12ga). Best of luck.
https://www.academy.com/shop/pdp/mos...action-shotgun