Why is Adleman's molecular algorithm for Hamiltonian Path linear?
You are right that in the coding of the graph one needs a possibly quadratic number of strands to represent the edges. 

My guess is that Adleman likes to distinguish the two kinds of operations. First there is the coding of the input graph, which may be very complicated to avoid accidental combinations. Then there is the number of laboratory steps to actually check the presence of Hamiltonian paths. 

I think this might be explained by the way he envisions how molecular computing can be used. Given a big "database" of available strands one performs a number of laboratory steps to find the solution to a  problem. This is then the same operation performed in parallel to zillions of molecules. This I read from the way how he explains the number of operations performed per second.