Ice dams are a heat problem, not a roofing problem. Warm air leaking from your living space into the attic heats the upper roof deck. Snow melts there, runs down to the unheated eave overhang, and refreezes. Meltwater then pools behind that ice ridge — and standing water on a shingle roof will find its way through, because shingles are designed to shed running water, not hold a pond.
The fix therefore has three parts, and only one involves the roof itself: sealing air leaks from the living space, bringing attic insulation up to current levels, and ensuring balanced intake (soffit) and exhaust (ridge) ventilation so the deck stays cold and uniform.