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We propose a class of photonic Floquet topological insulators based on staggered helical lattices and an efficient numerical method for calculating their Floquet band structure. The lattices support anomalous Floquet topological insulator phases with vanishing Chern number and tunable topological transitions. At the critical point of the topological transition, the band structure hosts a single unpaired Dirac cone, which yields a variety of unusual transport effects: a discrete analogue of conical diffraction, weak antilocalization not limited by intervalley scattering, and suppression of Anderson localization. Unlike previous designs, the effective gauge field strength can be controlled via lattice parameters such as the interhelix distance, significantly reducing radiative losses and enabling applications such as switchable topological waveguiding.