No matter what you do your going to leave big holes in hides with the .30-06. Your best bet might be to load heaver bullets and not push them to fast and hope you don`t hit a major bone. Your 12 ga. with #4 buckshot would be better if you can get within 30 yrds. of them.
I don`t know if you can still get them, but they used to make a sabot for the .30 rounds that let you fire .22 cal. bullet in the .30 cal. cases. They never caught on. You could get super verosities with them, but accuracy went to crap past abt. 75 yrds. In your case though, 75 yrds. would double your effective range, and in most guns you could take coyote to abt. 100 yrds.