A steering tool configured for making azimuthal and non-azimuthal formation evaluation measurements is disclosed. In one embodiment a rotary steerable tool includes at least one formation evaluation sensor deployed in the steering tool housing. The steering tool may include, for example, first and second circumferentially opposed formation evaluation sensors or first, second, and third formation evaluation sensors, each of which is radially offset and circumferentially aligned with a corresponding one of the steering tool blades. The invention further includes methods for geosteering in which a rotation rate of the steering tool housing in the borehole (and therefore the rotation rate of the formation evaluation sensors) is controlled. Steering decisions may be made utilizing the formation evaluation measurements and/or derived borehole images.