There was definitely slavery on Dragonstone, and likely at Driftmark as well:
“The Targaryens were of pure Valyrian blood, dragonlords of ancient lineage. Twelve years before the Doom of Valyria (114 BC), Aenar Targaryen sold his holdings in the Freehold and the Lands of the Long Summer and moved with all his wives, wealth, slaves, dragons, siblings, kin, and children to Dragonstone, a bleak island citadel beneath a smoking mountain in the narrow sea.”
In terms of when it was abolished…it seems to have happened fairly early on, because the description of the practice of first night on Dragonstone in P&Q wouldn’t make sense.