The fact that one of your state variables converges (settles) to a fixed value, is itself a sign that this state is stable, in some sense: if it were unstable, then the flow would have never brought you there.
It seems most likely that not only the state variable you're interested in converges to this fixed value, but that all the other state variables converge to a fixed value as well. It's worthwhile to check this, though: it could be that some other state variables are converging to an oscillatory motion. From the dynamical systems point of view, the former situation would indicate the presence of a stable equilibrium or stable fixed point of the system; the latter situation would correspond to a stable limit cycle.
On the amplitude of the 'test' perturbation: strictly speaking, if a state (equilibrium, limit cycle) is not stable, then any perturbation, however small, will not decrease in time. However, there is a difference between being not stable and being unstable. Take for example the pendulum, hanging down without swinging. If you give it a push, however small, the pendulum will start to swing, with a very small amplitude. Obviously, in the numerical setting, such oscillatory solutions with extremely small amplitude are very hard to discover, especially when the oscillation amplitude is less than the grid spacing, for example.
A steady state is stable if any arbitrarily small perturbation of that state decreases in time. That means that, if you want to test for stability, you indeed have to perturb all the state variables. Again, from the fact that the system seems to settle in this state, it's already quite likely that this state is stable, as the flow is (locally) towards this state.
Lastly, even though you only have to check stability against arbitrarily small perturbations, it can be very useful from the practical point of view to try larger perturbations, i.e. to test the robustness of the steady state you're interested in. Keep in mind that the system might (actually, will) be more sensitive to perturbations in certain state variables, and less sensitive to perturbations in others. This also gives very valuable information on the behaviour of the system 'around' the steady state.