Is the Sigma 18-200 F3.5-6.3 DC Macro lens compatible with the full frame D610?
No, it is not. The DC designation on Sigma lenses means it's a DX lens. Their DG designation indicates an FX lens. Nearly all zoom lenses that start with 16-18mm at the wide end are crop lenses. Full frame walkaround zooms more typically start at 24mm or 28mm. The Sigma 28-300 DG lens is probably what you're looking for.