Dragons do have their "don't stand so close to me" aura rings (pyroclastics/incineration point, etc) when they first enter combat.
But once they fly away and land again, the rings don't reappear.
Here's an ice dragon who has been on the ground again long enough to cast Soul Tear (and his wind wall), but there's no white ring around the dragon:
Here's an earth dragon who has been on the ground again long enough to cast Earth Spikes, but there's no red ring around the dragon: