How can I lock character movement to an object's axis?

you can use CylindricalConstraints to only allow an object to move on one axis https://developer.roblox.com/en-us/api-reference/class/CylindricalConstraint

as for actually figuring out how the Attachment0 should be rotated to get said axis, you can use the cross product of the surface normal and Vector3.new(0, 1, 0)

cross product will give you a vector which is perpendicular to both of the vectors you gave it; there are two possible vectors which can be perpendicular, so which one it gives you is dependent on the "right hand rule"

local function EnableCoverConstraint(surfacePos, surfaceNormal)
	local upVector = Vector3.new(0, 1, 0)
	local rightVector = upVector:Cross(surfaceNormal)
	local backVector = rightVector:Cross(upVector) -- all three "vectors" of a cframe have to be perpendicular to each other, so we need to get a new "back vector"
	-- CFrame.fromMatrix uses a back vector instead of a lookvector so. Yeah

	-- the rest of this code assumes that you just have a CylindricalConstraint parented to the HumanoidRootPart at all times, making a new one every time is kind of nasty when you can just disable it when it's not in use
	-- these are just example names btw, you can name the Stuff whatever you like
	HumanoidRootPart.CoverAttachment0.WorldCFrame = CFrame.fromMatrix(surfacePos + surfaceNormal, -- adding the surface normal pushes it away from the wall a bit, so your character isn't literally right next to it
rightVector, upVector, backVector)
	HumanoidRootPart.CoverAttachment1.WorldPosition = surfacePos + surfaceNormal
	HumanoidRootPart.CoverConstraint.Enabled = true
end

i haven't used cylindrical constraints before like, ever but i think you'll have to set the InclinationAngle to either 90 or -90