When you say it's going below the flapper, do you mean that the water literally drops to a point below any portion of the rubber flapper, that if you opened the flapper not a drop would pass into the flush valve? When you changed what you call the fill tube assembly, does that mean you changed out the entire flush valve and flapper, i.e. removing the tank, removing the tank-to-bowl gasket, undoing the nut, and removing and replacing the whole valve? If so, my only thoughts would be: (1) when you run your finger over the hole where the flush valve goes, is the surface nice and even and smooth and there are no rough spots or cracks in the porcelain? I have seen the Loves suggest a light sanding of the bolt holes, so perhaps a light sanding of this area would be prudent before trying your third flush valve assembly; (2) if you get another flush valve, I have had great success with the Korky 4030PK flush valve kit with tank-to-bowl gasket and hardware. You get their nifty valve with extra-large, twist and slide to right height overflow riser [no cutting necessary], a Korky plus flapper, and decent hardware. They also give you enough hardware to double-nut the tank-to-bowl connection (i.e. mount bolt and rubber washer on inside of tank and hex nut and metal washer on outside, then insert that bolt into bowl and put another rubber washer, another metal washer and another hex nut on the underside of the bowl, which is the superior way of mounting the tank on the bowl). If you center that valve in the hole, it has a big thick-ish gasket that puffs when compressed and sealed up nicely in the (now 3) toilets in which I have used it. I know this may seem elementary, but just be sure that you actually read the instructions and make sure that the right hardware goes on the correct side (i.e. inside or outside) of the tank.
If the water is actually going into the bowl, and it is not siphoning because the refill hose is stuck down into the overflow riser, and it isn't escaping around the flapper and it the chain isn't caught under the flapper, and the flush valve assembly isn't cracked and the china isn't cracked, then I don't see how the water is getting into the bowl.
For what it's worth, the Korky instructions are so clear and provide such a good illustrated anatomy lesson for the toilet and its operation, it's worth reviewing them and mentally checking against what you have done in your past installations... Korky Flush Valve Kit Installation Instructions PS Step 9 in those instructions is a little outdated; Korky will currently give you a clip that stands the refill hose above the lip of the overflow riser, to ensure there is no siphon created by the refill tube dropping below the water line inside the overflow riser (a very common cause of water flowing from the tank to bowl); it is properly shown in the diagram at the top of the instructions.