I'm not familiar with your model 4rnr drivetrain, so bear with me here...
If it has a lockable center differential, you would be able to leave it in 4WD (actually AWD) all the time with little effect except some tire wear and gas mileage decrease. Some Land Cruisers use that system, some of the Jeeps (Grand Cherokees, IIRC), various Land Rovers, and I think the Sequoia as well...not sure about any of the Tundra model years.
With those vehicles, you can use 2WD or AWD, and you can lock the center differential which gives you 4WD.
With 4WD, the front and rear axles cannot turn independently, causing the scrubbing and bouncing you felt. The lift should be fine

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There was a great thread, maybe in 1st-gen, maybe in the Tundra forum, very recent, about how and when people use their 4WD. There were quite a few comments about icy roads and foul weather. Basically, avoid making sharp turns on dry pavement in 4WD...dry also includes rain and mixed snow/ice/dry conditions, like melt after a snowstorm. In a straight line, you're generally OK. Check out that other thread...
-Sean