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Melee combatants may seem disadvantaged when battling a dragon. Dragons can fly and breathe fire as you stand around helpless on the ground. Sure, you could pull your bow out and attempt to do some damage with your Archery skill of 15, but I'm more likely to end up wasting arrows instead of hitting the dragon.
The ideal loadout for this strategy is the sword (axe/mace) and shield combo, as well as the healing spell in your favorites list. Having Unrelenting Force (even just one level of it) helps as well.
While the dragon is in flight, do your best to lure it to an open field (this isn't necessary, but it helps). When the dragon inevitably uses its fire(ice) breath, switch to your healing spell and use it to patch some of the damage done. If your health gets dangerously low, use a health potion if possible.
The dragon will eventually land, and this is where the melee combatant truly shines. Get to the dragon (preferrably without sprinting, you'll see why in a moment) and begin to hack away with your primary weapon. Block against its bite attacks and when it begins to use its breath, immediately bash it with your shield or use Unrelenting Force to stop it mid-shout. Strafe left and right to avoid the mouth, but avoid the tail because it can deal equal damage.
Unless you deal immense damage, the dragon will eventually begin flying again. Take advantage of this time to heal yourself with either healing spells or potions. Wait until it lands again, and then repeat this process.
This strategy has led me to victory against over 20 dragons, and it should help anyone struggling against dragons.