I would still finish the paddle, but next time, get clear vertical grain (CVG) wood. For best strength, the grain should be parallel to the SHORT side of the blank. Viewed from the end, the grain will look like this:

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...or something reasonably close. It's also important that the grain is relatively straight from one end of the blank to the other. I've made paddles with a bit of runout in the blades, but runout is not acceptable in the loom area. It seems to me that careful wood selection make all the difference in the world. I'm really picky I've yet to have any of my cedar paddles break.

Cedar, mahogany and Fir are typically available in straight-grained CVG form. Of the three, Cedar is the lightest and is easy to work. Mahogany is heavy, but a joy to work with. Fir is heavy, brittle and prone to splintering and tearout, and generally a pain to work with compared to the other two. If you happen to stumble upon some CVG pine, it will make a nice paddle. It's heavier than cedar, but lighter than Mahogany or fir.

BTW, if you cannot find good 2x4's, look at 4x4's. If you find one with good grain, you can resaw it into two paddle blanks. When all else fails, you can always laminate a blank from a couple of 1x4's on any other pieces that result in a usable blank.