I'm going to assume that your "leveling kit" was just a spacer. That's where the issue is.
Leveling kits, and lift kits that aren't the drop bracket type, DO NOT clear larger tires. Your suspension will still cycle through the same range and cause the same clearance issues as stock ride height.
Drop bracket kits are not a good solution. They're fragile, don't increase wheel travel, and induce bad handling characteristics.
What you want are quality aftermarket coilovers, aftermarket UCA, and a front alignment that provides 3.5° of caster or more. Then, start trimming. 37s are an easy fit on a Tundra.
In the rear, I'd recommend Icon RXT springs, extended brake lines, and a shock relocate to accommodate a 14" shock. Even if you don't relocate, a good 2.5" monotube will be a game changer.
In summary:
-Clearing bigger tires:
1. 3.5° or more of caster (pushes tire forward)
2. Trim all the unnecessary stuff out of the way (cab mounts, sheet metal, fenders)
-Suspension Performance:
1. Quality coilovers
2. Quality UCA
3. Aftermarket leaf pack
4. New rear shocks (ideally relocated for max travel)
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