I get this sort of problem with my joystick. I find that after i take a late / sharp turn at T1 on blackwood, all down the straight it wants to play on the grass.
My solution was that everytime LFS runs it required me to move all my axis about to get the centres central (something calibration in options) if i started the race, slightly turn to get to T1 then a sharp right it isnt calibrated right.
Try starting every race with a full turn left then full turn right, followed by full gas then brake. Got mine going a treat.