Go for flightgear
Having spent all too much of my (supposedly free) time playing with such things, a quick comparison of x-plane and flightgear would be:
X-plane allows you to "draw" the aircraft, and works out the dynamics from it.
flightgear has a couple of main flight-models - one which is table driven (called JSBsim), and one which works out aerodynamic characteristics, given performance figures (yasim).
If you want to demonstrate flying qualities by being able to vary the stability in a particular axis, I'd go for flightgear, and tweak an existing JSBsim aircraft in it.
As a specific example, the c172r model in flightgear has pitch co-efficients called things like:
pitch_moment_due_to_alpha = -1.8
pitch_moment_due_to_pitch_rate = -12.4
pitch_moment_due_to_alpha_rate = -5.2
Bringing these co-efficients closer to zero would (I think!!) reduce pitch stability, which you could then see in the simulator.
I'm not an aerodynamicist, but I hope I've helped!
Cheers,
grob103