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In this work, we theoretically analyze a gold (Au) core within silver (Ag) shell (Au@Ag) nanorod dimer biosensor for label-free molecule detection. The incident light on an Au@Ag nanorod strongly couples to localized surface plasmon modes, especially around the tip region. The field enhancement around the tip of a nanorod or between the tips of two longitudinally aligned nanorods as in a dimer can be exploited for sensitive detection of biomolecules. We derive analytical expressions for the interactions of an Au@Ag nanorod dimer with the incident light. We also study the detail dynamics of an Au@Ag nanorod dimer with the incident light computationally using finite difference time domain (FDTD) technique when core-shell ratio, relative position of the nanorods, and angle of incidence of light change. We find that the results obtained using the developed analytical model match well with that obtained using FDTD simulations. Additionally, we investigate the sensitivity of the Au@Ag nanorod dimer, i.e., shift in the resonance wavelength, when a target biomolecule such as lysozyme (Lys), human serum albumin (HSA), anti-biotin (Abn), human catalase (CAT), and human fibrinogen (Fb) protein molecules are attached to the tips of the nanorods.