Are modern airliners still stressed-skin aircraft?
Yes, absolutely. The skin transfers shear stresses between the stringers or longerons which run lengthwise along the fuselage and transmit tensile or compressive stress. Cutouts and openings (for doors and windows) need special local reinforcements, so the reduced structure can still withstand the loads.

Equally, the wing skin is the main element for the transmission of torsional loads. In today's airliners, the spar is part of the skin, but the wing skin ahead and aft of it is also a part of the primary structure. 

Generally, stress follows stiffness. When a structure is loaded, it will deform, and those parts which withstand deformation most will carry most of the load. In the end, all parts which have similar stiffness will carry loads in proportion to their cross section. Since most of an airliner's structure is made from aluminum, it has the same stiffness.

See this question for one example clearly showing fuselage skin deformation.