So, two pickaxe handles or shovel handles will work, but be just as picky as you would with a board!!!!
The trick is that the grain needs to run straight, end to end, in both directions. The rings are easy to see, so start there.
On any handle, look at one end. Pick the middle growth ring, and follow it visually from one end to the other; from grip end to head end, or whatever. That ring MUST stay basically in the middle, all the way down. If you split it in half it would follow that ring all the way from end to end. Got it?
But, here is the kicker! Look at the end again, and pick that middle growth ring. If you split the pick handle again, in your mind, exactly 90 degrees to that middle growth ring, it MUST still split into two almost exactly equal halves lengthwise. Get it? The PROPER stave for you could be split lengthwise no matter how you split it, no matter the angle, or the orientation of rings vs the splitting wedge.
Both the GRAIN and the RINGS have to run long enough to make a bow limb. Using two shovel handles, it can curl a bit or run off at the very tips, because you can cut several inches off the ends. But, be picky, picky picky. The last one I made, I went to three different stores and ended up with a pitchfork handle and a posthole digger handle. Get something with the nice thick bulge from middle to head end. You need that width.
Now, if the grain is straight in all directions, ring orientation doesn't matter too much, BUT, you are doing yourself a favor if you choose pick handles that have similar ring orientation. Pick handles have oval cross sections right? So, buy two that both have the rings running either longwise in the oval, or exactly crosswise to the oval. Almost like it is a flat sawn or a quarter sawn board. You want to take your bow limbs out of the widest part of the oval.
Shovel handles are straight and round, so you can rotate them to match up, but be super picky about the grain. Just because the rings run the length doesn't mean the grain does. Cut off the parts you can't use and decide how you are going to splice the handle. I recommend a W splice on a pick handle, but a shovel handle will be easier if you do a "pipe splice" by socketing them into a section of round or square stock, maybe with a diagonal cut so they match up deep in the socket. A straight front to back angled splice that good, too, but it should be pinned, doweled, or screwed, and then glued, and finally wrapped with wire, strong cord, fiberglass tape, rawhide, etc. The splice MUST stay wellell away from the fades like a 4" splice and a 9" fade to fade riser. It needs to begin and end in the thickest part of the handle.
Finally, if you get lucky, you will find two handles that also weigh about the same, and have similar growth ring thickness. Chasing a ring on ash is so easy it is almost fun, but just go down far enough to remove the roundness and the already cut through rings. Don't go halfway down into the stave unless you have to.
Now, caveats. Cheap shovel handles are much more likely to have grain run off that is harder to see, so by the time you find and pay for two suitable handles, you may have been better off just buying a stave or driving far enough to cut one. You are going to spend a minimum of $16.00 in my neck of the woods, and it could be $15.00 per handle. The small technical aspects of this are the hardest part; stuff like getting a good splice (which can take a lot of practice in and of itself) and navigating the shaping of a handle and fades around two long not-yet-spliced pieces. Think ahead. It hurts to ruin expensive, hard to find wood.
Also, all hickory is tough, but if you end up with all heartwood in one limb and all sapwood in the other, or one limb with four rings and the other with twenty, you may have some disparity between limbs, regarding hardness and weight. Likewise, if the ash has a lot of difference in the ratios of spring to winter in the growth rings, the woods will act differently, and you don't have the luxury of widening the stave to compensate. I would put the weaker wood on the top limb and make it longer, with positive tiller, if you can determine that ahead of time.
So, that's everything I can help you with. Good luck. Keep exploring options to find a better source of wood, and come back if you have any more specific questions.