Are businesses and individuals buying clean / green / renewable energy contribute to slowing down the climate change?
Your intuition is correct, actually switching from dirty good to clean good will also increase demand for clean good so price there might increase as well. However, this does not mean doing some voluntary change could not help as it also depends on how the quantity demanded changes with price (elasticity of demand). So switching is not necessarily unhelpful. 

However, at the same time it is also only a partial solution. In order to fully  internalize the externality it’s necessary to apply Piguvian tax to the dirty good at a level that fully internalizes the externality to people consuming the good.