223 can be fired in a 5.56 but 5.56 may cause issues in a 223. 5.56 chambers have a longer throat than the 223 chamber and the 5.56 is rated at a higher pressure by SAMMI (60,000 cup vs 50,000 cup)
The opposite is true for the 308 and 7.62. 7.62 can be safely fired in a 308 but 308 should not be fired in a 7.62. SAMMI pressures for the 308 (62,000 psi) are higher than those for the 7.62 (50,000 psi)
Shoot it! don't believe the hype. Check the primer for pressure sign, and do a chrono if you aren't sure. Most modern rifles are built with WAYYY more strength than what is needed. You'll be fine. Pull the heads and put a soft nose in it and go hunting.