Let's look at the geometry behind moving the ball forward.
Most likely you will be closer to lowpoint, or forward of it. Which means that you will hit the ball less in-out or more out-in, depending on your swing plane angle. Normally the clubface will start to close through lowpoint.
I suspect you moved the ball back when you were hitting push draws, as it is a result of the club moving in-out with a slightly open clubface to the target. If you move the ball more forward, you will hit the ball on lowpoint or slightly forward, which will be closer to a straight shot or small fade.
The part that can be confusing in all this is the swing plane line. Just because you align parallell to the target doesn't mean you'll swing along that line. Someone swinging a lot from in-to-out will have a hard time hitting a fade. If you swing perfectly square to the target line, moving the ball forward or backward can result in fade and draw.
Regardless of where your swing plane line is, moving the ball back will cause you to hit more out on it or less in and moving it forward will cause you to hit more in or less out. So to answer the question, yes, moving the ball forward of a position you hit a push-draw from can result in a fade. Here is a thread that discusses the exact topic: Hitting Up or Down with the Driver in an Inline Pattern
The clubface angle will have to be different on each ball position, but it usually is all by itself since the clubface moves from open to closed during the downswing.
I have done exactly the same as you with my driver. Once I started swinging more in-out, I started drawing the ball. But I prefer hitting a fade with the driver, so I moved the ball to the inside of my left heel. If I want to hit a straight drive or draw, I'll move the ball a bit back in my stance and push the hips more forward. If I get the clubface angle right, the result is usually good.
If you hit the ball on lowpoint with a square swing plane line and square clubface, it will go perfectly straight. Depending on your swing plane line, you may have to hit it on some side of lowpoint to hit it straight.
Btw, a push is a shot going straight to the right of the target, which is only possible with the clubhead moving in-out at impact and an open clubface (square to the swing path). Of course if you hit the ball with an open clubface on a forward ball position, the ball will start right and curve too far right.