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We have explored a concept for an advanced Normal-Conducting Radio-Frequency (NCRF) C-band linear accelerator (linac) structure to achieve a high gradient, high power e$^+$e$^-$ linear collider in the TeV class. This design study represents the first comprehensive investigation for an emerging class of distributed coupling accelerator topology exploring nominal cavity geometries, frequency and temperature of operation. The structure features internal manifolds for distributing RF power separately to each cell, permitting the full structure geometry to be designed for high shunt impedance and low breakdown. Optimized within operational constraints, we find that it is advantageous for the structure to be cooled directly by liquid nitrogen (LN), further increasing the shunt impedance. A crucial part of this design process has been cost optimization, which is largely driven by the cost of peak RF power. The first operation of a distributed coupling structure at cryogenic temperatures and the nominal operating gradient 120 MeV/m is also presented, demonstrating the feasibility of achieving high-gradient performance with a cryogenically-cooled normal-conducting accelerating structure.