A few ideas:
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You may want to try disabling steam input for the game, and just using the game's built in controller support. Go to the game's page in your steam library, hit the gear, go to properties>controller, and then pick disable steam input. Then try the game and see if it works any better.
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On PC if the game supports a walk keybind, with steam input enabled you could bind that to a back button, and then hold it down to walk.
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You said you tweaked deadzones, but was that in the game's settings or in the steam input settings? You may want to try editing the other one (ie lowering the deadzone size in steam controller settings, adjusting the deadzone size/type in game, and adjusting the stick response curve in game)
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You can also check community control profiles, the top ones may have a work around to fix it, or there's probably one that's simulating mouse/keyboard that would bypass this issue completely.