How do changes in flight affect L/D max and the speed at which I must travel to maintain it?
The best way to fly a constant L/D max is to have an angle of attack indicator.  Otherwise you would be constantly re-calculating the best speed as you burned fuel and changed weight.  Because most most GA aircraft don't have AOA, and are not too dramatically affected by changes in weight, max range and max endurance are generally expressed as an airspeed.

For a constant L/D max AOA, airspeed would actually decrease as the plane got lighter.  Think about the fundamental forces acting on the aircraft - lift is produced by a combination of airspeed and AOA.  If AOA remains constant, less airspeed will be required to produce the lift needed to counter the weight as the airplane gets lighter.

The reverse is true of course, if you are flying a constant airspeed you would need to gradually trim nose down to a lesser AOA as the aircraft gets lighter to prevent it from climbing. 

Regarding speedbrakes, they add drag so if you want to speed up you would not extend the speed brakes.  Speedbrakes are used to slow down, or to maintain a given airspeed at a higher power setting to improve throttle response and go around capability (upon retraction) during approaches.