First, as stated in several posts above that is a common aerodynamic function of airplanes operating within the realm of subsonic convergent airflow. The machine, when "trimmed" is only trimmed for the airspeed at which such "trim" is applied. Any deviation from that airspeed, deviation from balance (such as the Flt attendants walking up and down the aisle ), deviation from power setting, and a few other things will require an adjustment to trim. That is a fact, Jack!

For your specific model let's explore several items. 1. moving the Center of Gravity (CG) AFT as one so presented is absolutely the most wrong that can be attempted. On a wing such as your model has, the Center of Pressure (CP) will always be right in the vicinity of 25% of the Mean Aerodynamic Chord (MAC). Any CG aft will proportionally decrease stability of the machine. Therefore CG forward will proportionally increase stability. Any CG within the 25-30% of MAC will result in a nice stable aircraft that will maneuver well. Do not confuse the previous statement with certain applications of CG for specific desired functions that a learned pilot may desire. BTW, MAC can be used as the average chord in models because the difference is none to very little.

2. Adding/reducing power simply changes airflow over the control surfaces - changing trim - and changing airspeed which changes trim.
3. The chord line of the horizontal stabilizer is the real aerodynamic longitudinal reference, not the one drawn on a plan for reference. Therefore, if as offered above, you raise the wing's Trailing Edge, all you do is apply a negative wing incidence, which simply makes the horizontal stab the same as adding down elevator. The stab applies down force on the wing, such force being proportional to the square of the airspeed. Not a bad thing.

4. Now with a slow airplane such as a high wing trainer, flat bottom airfoil, the very best application which works quickly and efficiently is engine downthrust. You can experiment and find a down-thrust setting that will allow you to trim for level flight at your desired power setting and airspeed. Now when you add power, the engine will be pulling more down which helps offset the additional wing lift created by an airspeed/power increase. When you retard throttle for landing/slowing, the up-trim set for cruise (offsetting the down-thrust) will be less effective, thus the nose will not want to drop out of the sky and your machine will ease into the glide with less stick pressure required.

Therefore disregard all the fixes other than experimenting with downthrust to find a setting which you like. It may never be perfect, but it will allow far better flying than any else you can find. CG at 25 -28% edge of the chord back from wing Leading Edge and your problems will be minimal if any at all. Remember that most model airplanes have a moment arm of about 1 to 4, I unit forward of CG and 4 units aft of CG. I ounce in the tail needs 4 ounces in the nose to offset. That is too simple for full scale load planning but close enough for toy airplanes.
I suggest tapered plywood behind the engine mounting rather than washers which will eventually tighten into the wood and cause engine loosening.
Good luck.