Mr Pond,Perhaps you are unaware,but if you go here:
http://www.hornady.com/ballistics-resource/ballistics-calculator
Hornady will happily let you use the calculator,too!
Your .223 will have a lower ballistic coefficient than your .308....given the ammo you are shooting.That means the .223 bullets will lose velocity quicker.
That's why they are hitting lower.
Play with the calculator,and decide what you want in a trajectory/zero.The close up zeros,like 25 or 50 yds,IMO,are useful to get on paper,but I always fine tune at the longer range.You get to decide how much mid range trajectory you want on top of your line of sight.If you want a 300 yd zero,try about 4.5 in high at 100,and 5.5 in high at 200.Myself,I can work with that,I just hold low at 100and 200.The .55 gr bullet has a BC of only about .265.A 69 to 75 gr bullet will retain velocity better.If you want to shoot at 300,I'd try those.You will lose some initial velocity,so the 300 yd trajectory will be similar,but wind effect will be less,and it will make more difference at longer range.
If your .223 ammo is commercial Mil-spec,that will contribute to larger groups.its not known for precision.
Your rifle has a free floated barrel,yes?I forget .If it does,even with a chrome barrel,I would expect even a mediocre "lemon" to shoot insde 3 MOA at 300 meters,or 10 in.Actually,with good handloads,or match to varmint grade ammo,and a quality free floated rifle ,I'd be surprised if the rifle would not shoot better than 2 MOA at 300.I would not be surprised if it shot closer to 1 MOA.