ALO, you are describing the symptoms of a tail heavy plane. It's surprising, because manufacturers usually specify a very conservative balance point, so make sure you are measuring it correctly.

Tail heavy and a diving tendency are not contradictory at all. A tail heavy plane is unstable in pitch, so it is almost impossible to trim the elevator to a happy medium that will result in level flight. A little down, and the flight path will deviate down more and more; a little up, and it is the opposite. It makes a plane pretty hair raising and you have to fight it every moment.

The quickest way to check for this is simply to move the balance point forward a little. It probably won't take much to make a big difference.

Jim