Don't save Overdrive. Use it!
The green cubes do two things. First, they give you score, which is obviously the whole point of the game. But secondly, they fill the other bar around your ship: the green one just under it. This is your Overdrive bar. You can’t use it unless it’s full, so wait for the nice lady to tell you ‘Overdrive Charged’ and then look for a suitable place to use it.
Overdrive is best used when the screen is absolutely full of enemies, so try to hit as many as possible when you use it. During overdrive, your ship is automatically invulnerable and destroys enemies on contact, but you can still boost if you see a large group of enemies the other side of the map. Green cubes absorbed during overdrive don’t refill the bar, so sometimes it’s better to destroy enemies as far away as the beam allows, then pick up the green cubes they leave when your ship returns to normal. Then do it all again! There's no penalty for using it, so the more you can do it, the better. Obviously use it wisely, but don't forget to use it.