The application of the wellhead suction anchor in the second production test of natural gas hydrates (NGHs) in the South China Sea (SCS) was met with success. This design incorporates a central conductor guide pipe, which distinguishes it from traditional suction foundations. However, this addition resulted in a relatively high penetration resistance and a shallower penetration depth at the self-weight penetration stage. To mitigate this issue, the current study proposes an optimized design where the end of the suction foundation is sharpened. The installation characteristics of the traditional suction foundation and new suction foundation during self-weight penetration into sand are studied through laboratory tests and theoretical analysis. The flat and sharpened bottom shapes are considered in the traditional and new suction models. The effects of the initial penetration velocity on the initial penetration depth and soil plug and impact cavity characteristics are systematically studied. The results show that the self-weight penetration depth of the foundation with a sharpened bottom is 44.5% deeper than that of the foundation with a flat bottom. There are cavities around the foundation at the self-weight penetration stage, and the penetration depth is overestimated by 15%–30%. Finally, a model for predicting the penetration depth of the new suction foundation is proposed.