You're mostly right, but missing a bit.
Assuming that the density is uniform the effect of increased gravitational force applies everywhere except the exact centre. To explain, the gravity cancels out at the centre because you'd have an equal force pulling in every direction. With a denser sphere of the same mass the component forces would be greater, but still cancel out. But if you move away from the centre the net force towards the centre would be stronger in the denser sphere, at any given distance from the centre.
Edited by PaulK, : No reason given.