NAIROBI, Kenya—Islamic State’s push to co-opt one of Africa’s deadliest jihadist movements has come with an attempt to present a softer face to potential recruits.

Over the past year, the jihadist group also known as ISIS and Daesh has launched a broad recruitment campaign across Somalia to pry foot soldiers and senior operatives from al-Shabaab, a two-decade-old insurgency allied with al Qaeda that has made it very clear they have no desire to switch franchises. Stung by battlefield losses to larger al-Shabaab forces, Islamic...