The knock against the partition is it sheds the front portion off as it travels through the animal. I don't know why it's a knock as that is exactly what it was designed to do. The intent is that the smaller rear portion with now less frontal surface area will continue to penetrate. Mostly from my experience it does this very well.
The TTSX is designed to expand creating something of an "X" but maintaining most of it's weight. This of course means retention of momentum, but with a larger surface area thus reducing speed more quickly and stopping the bullet sooner.
So who wins? Well from what I've read the bullets that hold their mass eventually win out. Since you're shooting a smaller caliber against fairly large animals, I'm more inclined to take the weight retaining bullets.
If you were shooting a .300 Win Mag with 180gr Partitions, I would not concern myself in the least. I've knocked down elk and a Shiras moose using my 7mm Rem Mag and 160gr Partitions with little problem. But in the end we're talking about a .25-06 and much lighter bullets, so I'm more inclined to shoot the weight retaining bullets.
On the sticks I prefer in medium calibers or smaller to leave my forward (left) hand off the rifle as much as possible. Just use a finger or two to help steady the rifle. Let the sticks do their job. So I kind of grab the sticks just below the notch using my forefinger as a horizontal steady. I try to shoot the same way off the bench too in order to minimize changes in point of impact.
No doubt in my mind a rifle can change it's point of impact with changes in how it's held/rested at the front. Other than that, just keep practicing. Don't be afraid to do your own thing that feels best for you. Some prefer 3 legged sticks, others two. Whatever works best for you is what you should do.