The results of a search for direct pair production of top squarks and for dark matter in events with two opposite-charge leptons (electrons or muons), jets and missing transverse momentum are reported, using 139 fb$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$ of integrated luminosity from proton-proton collisions at $s=13$ TeV, collected by the ATLAS detector at the Large Hadron Collider during Run 2 (2015-2018). This search considers the pair production of top squarks and is sensitive across a wide range of mass differences between the top squark and the lightest neutralino. Additionally, spin-0 mediator dark-matter models are considered, in which the mediator is produced in association with a pair of top quarks. The mediator subsequently decays to a pair of dark-matter particles. No significant excess of events is observed above the Standard Model background, and limits are set at 95% confidence level. The results exclude top squark masses up to about 1 TeV, and masses of the lightest neutralino up to about 500 GeV. Limits on dark-matter production are set for scalar (pseudoscalar) mediator masses up to about 250 (300) GeV.