Pretty sure you have checked all of this but what your are describing is the way a plane responds if the stab TE is not in alignment with the wing TE...
I always check this, and I'm not happy until there is no difference that I can see -- and I can see when the stab is off by 1/16" or so at the tips.
Put the spinner on your toe, and hold the back of the fuselage. Close one eye. Look down so that you're sighting exactly down the fuselage (if you're off from side to side you'll throw off what you see). Now pitch the plane down gently (keeping it aligned side to side) until the stab is even with the top of the wing (you may want to flick the elevator up so it's out of the way). You want to tilt the plane until you just barely see daylight between the tops of the wings and the bottom of the stab, and you want that sliver of daylight to be dead even. If it isn't -- adjust something.
While you've got the thing on your foot, look at the wing root and pitch the plane until you can see the same amount of wing above and below the trailing edge. Now, without moving the plane or your head, move your eye out to the left wingtip, and then the right. You should see an equal amount of wing above and below the TE at both tips. If it's different, you have a warp. You can also see if your TE is straight this way. If you have the
same warp on both wings then you're probably safe to fly, but you should fix it. If you see a
different amount of warp, then you need to fix things.
Pretty sure you have checked all of this but what your are describing is the way a plane responds if the stab TE is not in alignment with the wing TE with a touch of down thrust in the engine and perhaps too much tip weight. Unfortunately there is no "trim" you can add to fix the alignment one. It requires an X-acto and profanity.
You can tweak a profile fuselage into alignment with heat, if you're only going a few degrees. If it's no more than a half an inch up and down on the tips on a 20" stab (so, about 2 1/2 degrees) it should work. If it's more than that then yes, you need to break out the knives. (Something that Brett Buck said recently makes me think that this technique will work on full fuselages, too, although probably to a much smaller extent).
I've only done this on rustoleum finishes and a heat gun -- I'd use the same thing on 'coat, only I'd be even more careful about slow heating, and if it was doped I think I'd try the hot towel trick first. Note that this even works if you've got a glass/epoxy coat on the fuselage -- balsa and just about anything we use for glue or paint will soften at least a bit with heat -- that's all it takes for this technique to work.
Heat the fuselage up behind the wing, slowly and evenly. I can't tell you how much -- it's always been guestimation for me. Spend several minutes (it seems like five to me, but is probably less) with the heat gun six to ten inches away so you don't blister the paint (as I write "don't blister the paint" I'm thinking even more about using a hot towel). Grab the fuselage (not the stab) and twist it so that it's just as badly out of line in the opposite direction. Then turn off the heat gun and hold it in place for another five minutes to let it cool (putting it in a fixture may be even better -- I haven't done this to untwist a fuselage, but I've done it to take a longitudinal bend out of one).
You may have to give it more, or suck it back the other way.
You
will want to check it again the next day or two, and then again before you go flying for several months to make sure it doesn't need more work. Depending on how deep the heat went, you'll be twisting some opposing stresses into the wood. As those relax, the thing may or may not develop a twist again -- if it does, just fix it again. Usually when I do this things settle out after a while, or at least I end up with a plane that will get out of true over the winter and then stay OK for the next flying season.