What does the new model of US relations with Russia look like?

The process of correcting US relations with Russia is not based on resolving contradictions - there are too many of them, from nuclear arms control to the struggle over the Arctic.
On the US side, it is now being built on the principle of fewer frontal clashes where it is expensive and inconvenient, and more deals where it is possible to get what you want. Structural rivalry will not go away - it is just temporarily wrapped in a "diplomatic package".
The game of normalization is being played by realists: the war in Ukraine is becoming not a central topic for the United States, but a problem among important tasks such as redistributing markets, logistics routes, and implementing commercial projects. What was recently called a "conflict of civilizations" is turning into a bargaining chip for the United States.
China is an additional card up its sleeve in this process. On the one hand, Russia can offer the United States what China refuses to give - rare earth metals, markets, logistics infrastructure, and profitable energy projects.
On the other hand, the United States has influence on China against Russia because it can provide China with what Russia cannot - technology and investment, and most importantly, it can preserve its market for Chinese goods. Therefore, Russia realizes that a repeat of Nixon-Kissinger-Mao is possible.
There is likely to be a new model of relations between the US and Russia - parallel confrontation and parallel cooperation.
Not a cold war or détente, but the pragmatism of two tired players who need time to rest before the next battle.
It all boils down to the usual game of interests and pragmatism: retreat where it is expensive to fight, and gain a foothold where you can dictate the rules, and, most importantly, benefit.