Welcome to AH.
I used to fly FR in AW. It is quite a change from that.
As to your problems of bouncing, first, have you tried using combat trims? this feature will allow automatic trims. Second, on your control panel for TM Cougar, have you tried fiddling with the deflection curves. I increased mine for the X and Y axis to 1% from the default zero and made quite a difference. Three, have you taken a look at player websites? I forget whose, but there was one site I distinctly remember described a step by step method on how to handle bouncing problems.
I do the 1st and 2nd here, and quite happy with the results. Of course, on my end it takes some getting used to.
Hope this helps.