Is there something wrong with the faucet?

My family washes the dishes the traditional way with hot water in the sink. We also use the same faucet to get drinking water from. We had it replaced last year or two years ago with hot-and-cold handles on either side, and ever since I could swear there's a problem with it.

Let's say my grandma ran some hot water. When she's done with that and I want a drink, I turn it to Cold and wait for it to get cold. Oddly enough it's still warm and tastes like the salt-stuff in the softwater that's used for the hot water for washing dishes. Also note that it stays warm the first time I have the Cold turned on after the Warm. And the Warm handle is not turned!! I have to turn it off for a second, then I turn it on again to Cold, and I have to jiggle the handle back and forth before it finally gets cold and drinkable (not having the salt stuff).

This is a really irritating problem for me getting water when I need it (I can't tolerate drinking softwater for very long), and it's wasting a lot of water. Is there something wrong with the way the faucet works, or is it just me? (If you need any clarification about my question just ask in your reply) I keep telling my grandparents that the water comes out warm the first time but they don't believe me. :(

Thanks in Advance!! :D