The no-insulation (NI) technique improves the stability and defect-tolerance of high-temperature superconducting (HTS) coils by enabling current redistribution, thereby reducing the risk of quenching. NI–HTS coils are widely applied in DC systems such as high-field magnets and superconducting field coils for electric machines. However, the presence of turn-to-turn contact resistance makes current distribution uneven, rendering traditional simulation methods unsuitable. To address this, a finite element method (FEM) based on the T–A formulation is proposed. This model solves coupled equations for the magnetic vector potential (A) and current vector potential (T), incorporating turn-to-turn contact resistance and anisotropic conductivity. The thin-strip approximation simplifies second-generation HTS materials as one-dimensional conductors, and a homogenization technique further reduces computational time by averaging the properties between turns, although it may limit the resolution of localized inter-turn effects. To verify the model’s accuracy, simulation results are compared against the H formulation, distributed circuit network (DCN) model, and experimental data. The proposed T–A model accurately reproduces key transient characteristics, including magnetic field evolution and radial current distribution, in both circular and racetrack NI coils. These results confirm the model’s potential as an efficient and reliable tool for transient electromagnetic analysis of NI–HTS coils.