What rate of twist is your barrel?
I get decent accuracy from my AR at 600 yards with heavy bullets (I can keep them all in an 8 inch circle), its not a varmint rig...just a plain old Smith & Wesson M&P 15 with a 3-9x40 Leupold scope on it.
It is pure destruction for coyotes at 300 yards with those 50 grain Nosler Ballistic Tips launched at 3,100 fps (.75" groups @ 100 yards)
1 in 9 twist barrel and it shoots REALLY good with 62 - 70 grain bullets.
Only 1 model of Remington 700 in 223 has a fast twist (1 in 9) barrel...I don't remeber the exact name of it, but its the tactical rifle (20 inch bull barrel).
If you have a 1 in 12 twist you will have better luck at 300 yards and less...because it won't stabilize the heavy bullets needed to reach way out there.