On 3 (Liquid-metal/T-1000) you actually can make things move properly. If the liquid can adjust its cohesiveness and strength (which it would need in order to change shape), then it can do this selectively i.e. it can form dense cores (bones), and elastic contracting ligaments (muscles). So by mimicking the interior structure of things, it can mimick their movements.
Justin, I have some questions here: What is the nature of the shape shifting? Is it a person who can morph into other people? A single animal? Multiple animals? Can they morph into non-biological objects? Complex non-biological objects with moving parts like axles or hinges? Is the change at-will, or in response to some environmental stimulus?
I think the liquid-metal is the most viable candidate for building a shapeshifter from scratch in a laboratory. Its the easiest way to attain the greatest performance and variety of transformation. A distributed computing system could manage the transformation and adjust various properties of the liquid to get the desired hardness, density, electrical conductivity, etc.
If the shapeshifter is an existing biological organism which has been modified to transform, then the choices become more limited. DNA I think is a limited, since it requires so much time and energy to grow into another form. You also have to worry about incompatibility between forms. For instance, if your blood was instantly replaced with a different blood type, it would probably kill you. You also have to worry about mass conservation, etc.
How about a parasitic/symbiotic liquid which is injected between cells? The symbiont could function like the liquid-metal shapeshifter, except that it would manipulate the host's cells along with its own structure. It could tap into the host's nervous system so that the host could think "turn into a X" and the symbiont could carry out the thought. Basically, image T-1000 inserted into your body and conducting micro-surgery on you. Of course, there are massive medical implications you'd have to account for - every shapeshifter would be a walking Emergency Room.