i'm not electrical expert but it seems like that is the issue. Your unit is like plugging in an aftermarket charger which makes your phone go haywire. Trade in that dock.
In theory it isn't an aftermarket charger since OP said he is using the charger that came with the Droid... but it sure is acting like it is.
The fact that it works when you "press" it more securely in the dock tells me... you got a bum dock and it is not connecting correctly. Return it for a new dock.
Sorry for bad wording, but i meant his OEM Dock is acting like an aftermarket charger making his Droid go crazy. My guess is some bad wiring within that Dock unit.
edit: nevermind bad cable. get that cable replaced! I wonder if my cable that came with the dock does that...I use the cable that came with the phone wallcharger.