Cutting a straight line is one of the most difficult things to do on a scroll saw. It is more difficult on solid wood pieces than ply because the blade tends to 'follow' the grain of the wood.
I find that I do best cutting straight areas by using slightly over sized scroll saw blades for those places. If you are typically using a 2 or a 3, try putting a 5 in and seeing if that helps. If it still pulls off center, go up one more size to a 7.
I think you will always need to angle the piece by varying degrees when cutting straight on. This is because of the nature of the blade which was described in the previous comment. That is why we can't really use a 'fence' or any type of wall to cut straight - the blades are too thin and subject to twisting and we constantly need the 'human factor' to correct our cutting line. (That is a good thing - we are not obsolete yet!
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On many of my candle tray patterns I have either a square or circular edge. Certainly the center piece that drops from the middle of the trays are either round or rounded squares. On all of my instructions, I advise to use larger scroll saw blades for the reasons explained above. It also helps to look about half an inch or so in FRONT of the blade while you are cutting, as it helps to steady your hand (It helps me, anyway).
I hope this helps you all out. The larger the blade, the more stable it will be and the less likely it will be to twist from the discrepancies of denseness in the wood grain. Keep that in mind and you may find you are more successful.
Sheila