Hello and Welcome -
My name is ***** ***** Im happy to help / advise.
You are getting the "Service Stability System" message - because the computer that has overal control of that system has detected a fault somewhere in that system.. and hence its giving you the lack of stability on the road as you describe.
To explain further -
The stability system is an addition to your ABS braking system. It operates with / via additional sensors - steering wheel position sensor and YAW sensor and is designed to determine if the vehicle is losing directional control... ie, forward movement -and aids to keeping it stable.
It works ( basically ) by apply brakes to specific wheels to attempt to correct if the vehicle is sliding or losing traction. If the stability system light is on, ( or the message you have ) then the system is disabled for a failure. It can be disabled for several reasons. If the abs light is NOT on as well then it is likely not a wheel speed sensor failure. It can be a steering wheel position sensor or Yaw sensor. It can also be disabled due to an engine control fault of some sort that will disable traction control. As I said above , the fault is somewhere in the system - one of the sensors mentioned above usually but you cannot guess at whats gone wrong - the only way forward now is to have the car connected to a suitable computer which will scan the computer for fault codes - which will be there because you have that fault - and the fault message ( as above ) on.
Once the codes are identified by the said machine , the fault then can be fixed properly. You will of course need garage / shop help for this - (Obviously the mechs you have taken the car to do not know what this is how or how to deal with it and / or don't have the proper equipment to scan the computer )... and being this is not as simple a system as say an engine management system which a lot of mechanics are used to deal with - these are not - I would strongly recommend you use a Chevy dealer for at least the diagnostics - and then fix from there.
Let me know how you go..and / or let me know if you need anything further on this..
~>>~~>>~ Meanwhile now - Please do click on the "Exc / Ok rating" for me at this time.. "Thank you" in advance for that .
.. Note that you can get back to me anytime at all after doing the rating - it doesn't close you out --
Best Rgds, MIKE.