Do I need to include instrument resolution in the uncertainty budget?

Include it unless the repeatability data clearly already expresses it. The standard component for digital resolution δ is a rectangular distribution over ±δ/2: u = δ/(2√3) ≈ 0.29·δ.

The double-counting question: if your 10 readings scatter across several counts, the dispersion already contains quantization and the resolution line is redundant — many labs still carry both when the resolution term is negligible, which is harmless. But if all 10 readings are IDENTICAL (scatter hidden below resolution), the Type A estimate is zero and misleading — then the resolution component is essential and is your floor.

Rule of thumb: u_resolution is a floor, u_repeatability is measured; take whichever treatment is honest for your data, and never report combined uncertainty below the resolution floor.