Just regarding riding straight lines going forward - I never, ever, ever use straight lines on a green horse or one on the forehand, unless it's to test my work on a curve. On a curve, you have control over bend, and with bend come softness making it SO much easier to engage a hind leg.
On the straight, it is REALLY easy for a horse to run itself onto the forehand even more - yes the stride may lengthen, but out of disbalance rather than engagement.
Totally cheesy but "Bend is your friend, straight is your horse's mate"! An international GP Dressage rider told me that at a clinic, and the saying has stuck ever since :P