They don't speed up over time. I'm going for a classic platformer style where the player automatically moves at their top speed with no acceleration/deceleration. How it works is that if the player starts moving, their velocity goes straight from zero to the walk speed or if they hold the run button, to the run speed. No acceleration, no speeding up over time. It's instant.
What's happening is that the player can also manipulate their speed while they're in the air which they shouldn't be able to do. If they're airborne, their speed should stay at either the walk speed or the run speed depending on what the player's previous state was the entire time they're in the air.
Physics process in case it helps:
func _physics_process(delta):
input();
velocity.y += gravity * delta;
if velocity.y > maxFallSpeed:
velocity.y = maxFallSpeed;
velocity = move_and_slide(velocity, Vector2.UP);
