In regards to Dain's folk...
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In battle they wielded heavy two-handed mattocks; but each of them had also a short broad sword at his side and a roundshield at his back.
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Tolkien never claimed all dwarves used axes, or even used them generally in battle, that is an accretion piled on Tolkien lore from elsewhere (like dwarves speaking in a Scot's accent, laddie). Mattocks are a miner's tool, and would be readily available to any dwarf (axes not being of much use in the type of subterranean manses dwarves engineered). Also, there is a tradition of dwarvish sword-making that goes back to Telchar of Nogrod.