I am using 5770 also and it is working fine.
First. I think there is no problem with the driver and the card, because it is showing a hash rate 380 Mh/s.
Second. What is the setting of your CGminer?
I'm using GUIMiner because I haven't figured out how to make CGminer to switch to my 5770
I just started up CGM, here's the log from it:
[2013-04-20 14:14:47] Started cgminer 2.11.4
[2013-04-20 14:14:47] Loaded configuration file cgminer.conf
[2013-04-20 14:14:47] ADL found less devices than opencl!
[2013-04-20 14:14:47] There is possibly more than one display attached to a GPU
[2013-04-20 14:14:47] Use the gpu map feature to reliably map OpenCL to ADL
[2013-04-20 14:14:47] WARNING: Number of OpenCL and ADL devices did not match!
[2013-04-20 14:14:47] Hardware monitoring may NOT match up with devices!
[2013-04-20 14:14:48] Probing for an alive pool
[2013-04-20 14:14:51] Long-polling activated for
http://pit.deepbit.net:8332/listenChannel [2013-04-20 14:14:51] Disabling extra threads due to dynamic mode.
[2013-04-20 14:14:51] Tune dynamic intensity with --gpu-dyninterval
[2013-04-20 14:14:54] Thread 1 being disabled
[2013-04-20 14:14:56] Accepted 2eac6b26 Diff 5/1 GPU 0
[2013-04-20 14:15:56] Accepted da7127d4 Diff 1/1 GPU 0
The 5770 is on platform 1, device 0
Edit: I've been able to get it mining with GUIMiner, but it will go for a little while, 30 shares or so, then it will stop mining.