See, capping it that way makes sense on one level, but on the other it only matters every time you roll - so the max-DEX Barbarian who rolled a 5 and a 9 that morning will never go before the guy who dumped DEX but got a natural 20.
Beyond that, I want this to plug into both characterization - the paranoid guy always thinks fast, if not necessarily well... - and into a caution-versus-speed tradeoff; go slowly to avoid traps and ambushes, or go quickly to get the drop on enemies?
To that end, I think I'd get rid of the d20 roll entirely. Default pace is +10, accelerated pace is +15, cautious is +5.
Then when you get into a fight, an 'eyeball it' modifier of +5 to -5 is applied based on how ready you for that fight. Once a Barbarian hits 7th level, their default for this modifier is 5 higher, but their maximum isn't increased.
DEX mod is factored in as normal.
Perception uses a very similar thing. You can roll it as a skill, generally for eavesdropping or a hurried search, but your Passive Perception is your total Perception modifier, +10 for a default pace, +5 for a hurried one, or +15 for a cautious one. Then add a +5 to -5 modifier based on how cluttered and obscured the area is, and state that Favored Terrain lets you treat your favored environments as always giving +5.
Sound good?