I believe this one is popular in the US (and is it Jenibelle has one?). Bottles are a total pain sometimes. I would rather have one drip a bit than not let enough water out, but yes any bedding that has got wet needs to be removed (just a handful maybe in that area until the wet area is gone - and replaced with a dry handful or two) or it will go mouldy. I also have a stone under the bottle - it doesn't matter if that gets wet. I wouldn't use a dish though as the water in that can get contaminated easily.
I think most bottles drip a bit - and if they don't drip some can not let water out easily, which is something I'm paranoid about too. For now just put something under the current drippy bottle until you get something else that works well (like the one below maybe). Sometimes if a bottle doesn't let water out well, leaving it very slightly unscrewed and only 2/3 filled can help the water come out easier. I tap my finger under it to check water is coming out easily. If you unscrew it too much, the water leaks out - it's finding the right level of tightness or looseness.
Amazon.com : Kordon/Oasis (Novalek) SOA80304 Bell Bottle and Hold Guard Small Animal Value Set, 4-Ounce, variety colors : Pet Water Bottles : Pet Supplies