The battery life of those cheap Walgreens AAA radios is excellent. Mine ran for something like 15 hours on a 750 mAH AAA NiMH charge (I did an overnight test). With alkalines it should be well over 24 hours, with lithiums even more. They use very little power because there's no speaker, just earbuds. They are really minimal radios though, FM-only (though stereo), no tuning dial (press a button and it scans to the next station). Mine has a volume control but some of them don't even have that.

The little earphone radio that I EDC runs on three button cells and might run for only a few hours. I should measure the current drain (to estimate from) or do a runtime test sometime.

I think the issue with your Sony radio isn't AA's vs C's but that it's using excessive power for some unknown reason (whether design or defect). A C cell holds about 2x as much energy as an AA. AA's are pretty much standard for everything now.