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Hi Lloyd,
Generally, you start adjusting the highest band and work down through the other bands but as, you discovered, with a multiband trap antenna there are interactions among the bands so tuning is tricky. Since it's easier to trim than to add wire, on the first pass try for resonance at lower than optimum frequencies and make note of how tuning a lower band affects the previously tuned higher band.
Although this antenna is a great idea, there is a flaw in the tuner design that might make it difficult to achieve low SWR on all bands. The tuner uses an autotransformer so the shield of the coax feed line is actually the counterpoise. That means the 10' counterpoise isn't a counterpoise at all because the feed line is. What you get isn't actually an EFHW antenna but an off center fed dipole. That makes tuning on three bands even more difficult.
The best way to tune an EFHW antenna, IMHO, is with a tank circuit coupled to a feedline with a secondary winding that isolates the feedline from the antenna. See <> for an example. For the QRP Guys antenna you probably need to tune the antenna with a very short feed line, say, 10'. Once you have all three bands in the ballpark, try a longer feed line to see how that affects the tuning. You might need to try different lengths of feed line to make the antenna work on all bands. An alternative is to modify the tuner to accommodate a true transformer rather than an autotransformer.
An antenna analyzer certainly makes the job easier because it tells you where the point of resonance is. Without it, you're confined to making SWR checks within the band. Perhaps you can borrow an analyzer from someone nearby.
Hope this helps.
73,
Pat Byers? ? VE3EUR