Phil...I don't have the Sabi handle but I do have one of the Cadet handles. I have a Gillette New (well, and Otto Roth, but same thing). The Roth has the wider space between the raised channel running down the center. Some of the news I know have very narrow openings where the handle threads in.
Even with the wider spacing I have, the Cadet handle is just a bit too wide to fit in between the grooves and sit flat on the head. The first few times I tried to tighten it down the head was sitting on there a bit crooked. However, I found that if I start the handle on the thread and then literally try to pull the head out of the threads as I screw down with the other hand, it will sit just on top of the raised groove on both sides. I tighten it down (not as hard as I can, but firmly) and the razor head is perfectly level. I have shaved with it twice this way with no issues.
I know Turtle has a Sabi II head and he uses it with all of his "free range" heads....heads he owns with no matching handle. I know he has a New in his collection. So, while I can't say for 100% certain, I am fairly confident you can use it with the New. Hopefully Turtle or someone else with both items will chime in.
Such a nice, heavy handle makes a bigger difference in the quality of shave you receive than I would have ever imagined. The extra weight is really special. I will never be able to go back to a regular, hollow, lightweight aluminum or resin handle.
My Sabi handle fits my Gillette SC head (thread wise). However, as others have mentioned, it does not fit into the indention between the raised center ridge on the bottom plate. I resolved that issue by finding a small rubber/nylon washer that fit over the screw and within the indentation. Now the Sabi sits atop the washer and fits well.