Help with making it give only 10 per touch?

I'm assuming your problem is that both legs are touching the part at the same time, and possibly walking on and off the part. Add some sort of debounce into your code like so:

local Part = script.Parent
local deb = false --set the variable

Part.Touched:connect(function(part)
if deb == false then --make sure it's false, so you don't spam the button
deb = true --set it to true so it won't run the code again until it's false
local player = game.Players:GetPlayerFromCharacter(part.Parent)

if part:IsA("Part") and (player) then
local points = game:GetService("PointsService")

if points:GetAwardablePoints() >= 0 then
points:AwardPoints(player.userId, 10)
wait()
script:remove()
wait(5)

end
end
wait(.5) --half a second before they can touch the part again for more points
deb = false --enabling the code again
end
end)

Hope I helped!