Low-Earth-Orbit (LEO) satellite constellations provide Internet connectivity across the globe, but their network design poses significant challenges due to the large number of satellites and their fast movement. Named Data Networking (NDN) can bring many benefits to LEO satellite networks such as data-centric security, scalable content distribution, and intelligent data plane. A key enabler to these benefits is NDN's stateful forwarding, which unfortunately can be disrupted by the frequent handovers between the satellites and the ground terminals. In this paper, we investigate the impacts of these handovers on NDN's packet delivery and propose effective mitigation mechanisms. Using a newly developed packet-level simulator and in-depth analysis of packet forwarding behavior during handovers, we show that consumer handovers and producer handovers both lead to temporary packet losses but for different causes. To mitigate these problems, we design a new Interest retransmission strategy to handle consumer handovers, and a new forwarding strategy to handle producer handovers. Our evaluation shows that these solutions are effective in reducing packet losses and delivery time. This work sheds insights on how NDN can work in the presence of frequent satellite-ground handovers to enable data-centric communication in this new network environment.