I always read it (likely wrongly) that although the dwarves weren't corrupted by the power of the rings like the Nine were, they did suffer some, and that's where the gold sickness originates from. But that's just my interpretation of canon. I do believe they never became Sauron's slaves, like the Nine were, but that they were impacted in other, less obvious, ways. After all, Thror was the last to rule with a ring, though Thrain had it after (he also had no gold, but he did rather stupidly decide to fight for Moria, so there's that - course, Balin does that too later on, so maybe Rings of Power don't play in here).