Interesting question. I suppose one should compare several scenarios
- Lie still
- Go forward - either straight, or hard to port, or hard to starboard
- Go in reverse
The rate at which water enters the ship is (to first order) proportional to the pressure differential - lower the pressure and live longer. Maybe even long enough for the Carpathia to come and rescue you...
If you think about the bow wave of a ship, its presence tells you that pressure builds up in front of the ship. Titanic cold not turn hard enough to port to pull a void on the starboard side, so I don't think any of option 2 is viable.
This leaves the question whether (3) would allow the bow to come (relatively) out of the water. I honestly think that once they realized they were surrounded by ice bergs, nobody would have much stomach for trying to go full steam astern (not very fast, with only two of the props turning the right way) in an attempt to change the equation of mass.
There were multiple small gashes far below the water line - where the rivets popped due to the stress of the impact. Based on investigations of the wreck, this is roughly what they might have looked like:
The draft of the ship was about 34 feet (10 meters), putting the deeper gashes about 6 meter or so below the water line. The fractional change in pressure that you could effect by any maneuver to raise the bow / change the profile of the water line would be at most 2 meter (being generous here), which in the very best case would have slowed the sinking by an hour. Tops. And in the process I am not sure what they might have run into, going backwards.
I hope someone has a hydrodynamic model to put a better confidence interval on my "two meters tops" number. I expect that it is very high.