Article discusses the effect of rising hardware prices on the deck.
Some highlights:
How much worse has the pricing situation gotten for Valve since November? Superdata Research founder and SuperJoost newsletter author Joost van Dreunen suggested that the 512GB Steam Machine model would probably run $50 to $75 more than he expected when the Steam Machine was announced, and to expect a price “potentially $100+ above target” for the high-end 2TB model. That would mean a $599 to $629 price at the low-end and $849 to $899 for the high-end model, in his estimation.
Wedbush Morgan analyst Michael Pachter agreed that, even with the additional component costs, Valve would likely “try to get it out at $599 or so for the 512GB version,” A starting price higher than that would mean “abysmal” sales, he added. “I think $700 is a death sentence and $1,000 is unsellable.”
I'd recommend reading the article though, it has a lot more of value than just those quotes. It goes on to talk about how the price increases will likely hurt valve more than traditional console makers, and how these increases will affect sales.
A few ideas:
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.
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.
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)
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.