Hello Aaron.
According to Trimmer Dillon the ship's engines stopped 1.5 minutes after impact and started going astern 2 minutes after impact. They ran astern for a further 2 minutes before they stopped again. When they stopped, you can be sure that the ship was almost dead in the water. The reason I say so is because the next engine order was an ahead movement. In other words, about 4 minutes after impact Titanic had come to a halt. There might have been a very slight way on her
If we fit the foregoing to the story of Lightoller, it seems that, if he saw Titanic still moving a head, then the engines were in the act if building astern power and Captain Smith was bringing his ship to a standstill... probably 3 minutes after impact.
To find out when she was just stopped and beginning to go astern, he would watch the propeller wash round the stern. Because of the flat calm conditions, he would see it as an ever-widening circle of churned-up water. He would have to judge the exact moment to ring down stop or the ship would start going astern again.
You talk about an underwater spur effecting the path of Titanic. Not as far fetched as one might think.
Rob is spot -on with his assessment; no way could Titanic have dragged such a large lump. Unless, of course, it wasn't as large as everyone thinks. A Growler, which can be like your overturning berg, can do a heck of a lot of damage and Titanic could have conceivable moved a very small one. But even then, by the rules of old Sir Isaac Newton, there would have been a resultant effect. We have other similar events to guide us.
When a ship accidentally hits a quay wall with her 'shoulder', her stern kicks out from the quay-side and, if there is sufficient impedance to forward travel, the bow momentarily cants toward the point of contact. As her near side moves back toward the quay, a cushion of compressed water acts like a temporary fender and hold her off before it spills aft and the ship comes back alongside.
We must be careful when considering stories about the colour of the iceberg. As you probably know colour is actually the shade of light rays reflected off an object. You cannot see an iceberg at night unless it reflects light shining on it. On a dark, moonless night, an iceberg will only be seen from a ship if it reflects light from the ship. blocks out light from another source or, as in the case with Captrain Rostron, reflects the light from a shooting star or comet streak. QM Olliver saw it as " a kind of a dark-blue. It was not white." he was seeing the reflected light from Titanic's accommodation and the green navigation light on the bridge wing as the ship passed he berg. By the way, that iceberg must have been outside the No.1 Emergency boat or below it's keel other wise it would have swept that boat away.
If anyone saw that iceberg astern of Titanic after she had stopped, then it was less than half a mile away on her starboard quarter. In that case, if Titanic was pointing north, the berg would have been to the SSE of her. There is no way the ship could have made such a tight reverse turn with her engines rapidly slowing down.
The idea of Titanic pointing north, is, in my humble opinion, a myth developed to fit the SS Californian to the lights seen on Titanic's port bow after, not before she stopped and to fit the orientation of Titanic's bow section on the sea bed.
If , as I believe, she had never turned north of due west, then the vessel seen on her port bow was most certainly not the Californian.