your vibration it could be that your tires need to be balanced and your car needs an alignment. While there, check all the suspension components. You can also change the steering shock, which is probably shot by now, if it is the original.
The "unsteady" feeling may be the rear suspension links.
As far as temperature, was the car doing this before you changed the coolant? Did you change the thermostat?
Do a search there is a lot of GOOD information on all these topics.