Can I give my damage phone to the apple store and get a new device with a better price?

Apple has a trade-in/recylce program but when I entered a water-damaged iPhone 6 into the "See what yours is worth" link in the following, it said it there was no monetary value. You may want to try an Apple Store to verify. http://www.apple.com/iphone/trade-up/

You cannot hand in a phone under the trade in program if it is damaged, since, as stated, those are considered to have zero monetary value to put towards a trade in.

You could pay the out of warranty exchange fee and get a replacement, but under the OoW program, you will get exactly the same color, storage and model as you trade in. There is no up-grading allowed under OoW replacement. You get a like new replacement though for far less than a new device.