As the CG is moved aft, the elevator will be more effective. Less stable? Not as much as more sensitive. If you adjust the elevator throw to match the sensitivity you won't see or feel any difference at all. That is, until the CG gets to the NP and behind.

As the CG is moved forward, the elevator and stab starts having to carry an artificially increased load. The farther forward the CG moves the more leverage that weight has to push the nose down. So the tail has to push down to carry it. And the extra artificial force winds up being doubled and winds up stealing the tail's effectiveness. So the elevator gets less sensitive. And most people think the airplane is getting more stable. Less steerable yes. More stable? Not appreciably.

The porpoising you describe isn't usually what you see from a tailheavy airplane. When a tailheavy starts to pitch what you usually see is your stick movement PITCHING the airplane QUICKLY and HARD whichever way you jam the TX stick. And even when you very gingerly move the TX elevator control, that very sensitive elevator THROWS the airplane too much. And then you see a crash 'cause you can't fly something with that sensitive an elevator.