This problem is resolved as of OFP version 1.30.
If you're running OFP version 1.20 or below, whether you're driving a land vehicle or piloting an aircraft in OFP, there's a bug in OFP that can make you lose control.
One solution is to hit your keyboard's ALT or asterisk ("*") key, which will put you into "mouse look" view but will force your vehicle to go straight. Hit the ALT or asterisk key again to quit "mouse look" view and resume driving.
Another solution to this problem is to jump into 3rd person view and, when the cursor returns to the center of the screen, return back to 1st person view.
Yet another solution is to quickly tap the "A" or "D" keys or right or left arrow keys.