From what you describe, I'm guessing that the new Freud blade you bought is
a thin-kerf model. So you end up with a 1/8" splitter trying to wedge
itself into a 1/16" saw kerf and it won't fit. The simple solution is to
swap the thin kerf blade for a full thickness one. Then the two sides of
your workpiece should slide neatly around the splitter.

Lee

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