I dont think it's a hardware problem.
I had this auto left-right oscillation problem during the straights with my Thrustmaster Rally Gt Pro wheel too.
Also, if you watch an AI driver performing a lap you 'll notice that while in a straight he does this thing exactly. He does not keep the wheel steady, he does this rapid left-right movement all over the straight.
(So I guess there's something wrong with the LFS engine here.
Also I don't have this problem in other games like RBR.)
The solution I came up with was to let the wheel take care of the FF control, (it has an option in the control panel) not the LFS and I reduced the FF in LFS from 80% down to 25%. Now it's working fine.
So, the solution I think is to keep messing up with the settings of the FF control until you find the right combination for yourself. It'll take some time but eventually you will sort it out, I hope.
Unless I'm totally wrong