Drillings have always interested me. I know the German Airforce used to use them for pilot survival guns. The ones I have handled are not light. And since the shotgun barrels are over the rifle, I am not sure how accurate you are likely to be with the rifle.
I decided to partly assuage my drilling lust and got the poor mans drilling. A Savage 24C.
.22 LR over 20 gauge. Very compact and light weight (5.5 lb) and this is probably THE ultimate EOTHAWKI gun. The rifle shoots great, the shotgun is definitely more for occasional use. I used it to shoot a couple of rounds of skeet and my shoulder told me I really shouldn't have. It would be fine for hunting (especially for the pot).
You can get them in larger calibers (.223 over 12 gauge is particularly popular). I have seen them in 30-30 over 12 gauge as well. These versions are a lot heavier than my 24C. I would hate to have to lug one of these bigger ones around all day.