Your dentist might have better advice. I know that we took our first off the bottle first, the binky second. What we did was slowly phase out milk in bottles but let her have water (we watered down her milk so it was a transition). Eventually it was her choice to quit drinking water, she'd rather have milk in a sippy. Find one she/he really likes. As for the binky, she was old enough to talk, around 1 1/2 I think. I snipped the very tip off every binky and didn't tell her. She thought they were broken and imagine that, all of them were broken! Three days later I snipped a bit more. And three days later a lot more. By then she was over the broken binkies. And again, it was all her decision so no tears!
With this baby I don't plan to take them away quite so early. I have heard with binkies that as long as you regulate it (like make them take it out to talk so that they develop speaking properly) then it is fine up to age 3. I am going to be more relaxed this time around.