I had the exact same problem when i first got my 6670.
It's usually caused by the Catalyst AI which is part of the ATI software that comes bundled with the driver.
The Application settings conflict with the driver settings, sometimes even when you set the driver to "use application settings".
Two things I've found that worked to fix this were:
1. Uninstall Catalyst Control Centre and the driver.
Reinstall just the diriver.
Install RadeonPro. This allows you to set configure graphics settings and turn off the Catalyst AI on a game by game basis.
2. (method I'm currently using)
In Catalyst Control Centre under tab for Gaming, set to the following:
Anti-Aliasing mode : Use application settings
Anti-Aliasing Samples : Use application settings
Filter : Standard
Anti-Aliasing Method : Multisampling
Anisotropic filtering mode : Use application settings
Anisotropic Filtering level : Use application settings
Texture filtering quality : High quality
Surface format optimization : On
Wait for vertical refresh : Off, unless application specifies (This is the one which causes most problems for screen tearing as it conflicts with in game settings when set to on)
OpenGL triple buffering : Off (this causes most of the lag when set to on)
Tesselation mode : Use application settings
Maximum tesselation level : Use application settings
Then it's just a case of tweaking in-game settings until you get it looking the way you like it.
Using these settings I've been able to run Crysis 2 on maximum graphics at 1920x1080 even with the Directx 11 patch. I can run it on high with the Hi-res texture pack as well.
My system specs:
Operating System: Windows 7 Professional 64-bit (6.1, Build 7601) Service Pack 1 (7601.win7sp1_gdr.130708-1532)
Language: English (Regional Setting: English)
System Manufacturer: Me
System Model: P5Q-VM
BIOS: BIOS Date: 11/04/08 12:33:35 Ver: 08.00.14
Processor: Intel(R) Core(TM)2 Quad CPU Q9550 @ 2.83GHz (4 CPUs), ~2.8GHz
Memory: 4096MB RAM
Available OS Memory: 4096MB RAM
Page File: 2887MB used, 9397MB available
Windows Dir: C:\Windows
DirectX Version: DirectX 11
DX Setup Parameters: Not found
User DPI Setting: Using System DPI
System DPI Setting: 96 DPI (100 percent)
DWM DPI Scaling: Disabled
DxDiag Version: 6.01.7601.17514 32bit UnicodU
Card name: AMD Radeon HD 6670
Manufacturer: Advanced Micro Devices, Inc.
Chip type: AMD Radeon Graphics Processor (0x6758)
DAC type: Internal DAC(400MHz)
Device Key: Enum\PCI\VEN_1002&DEV_6758&SUBSYS_E194174B&REV_00
Display Memory: 2807 MB
Dedicated Memory: 1015 MB
Shared Memory: 1791 MB
Current Mode: 1920 x 1080 (32 bit) (60Hz)
Monitor Name: LG 23EA63(Digital)
Hope this helps