Without offending the psionic-is-not-magic! crowd, the problem of converting high fantasy D&D to a darker sword and sorcery setting like Darksun isn't too much the ability to cast spells, or having spell slots; its the spell lists themselves.
Many D&D spells can easily be flavoured as psionic powers, and in the absence of a psion or a mystic class, classes like the bard, ranger and paladin (and arcane trickster and eldritch knight) readily serve as psionic user classes. Use the spell point variant for a more AD&D 2e psionic feel.
We started a Darksun game a while ago and I wanted to play a psionicist, and the bard provided a good platform for it. Take vicious mockery and rename "ego whip" and it already feels Darksun-y. Although we didn't get that far, you could easily cull and mix 5e D&D spells in a psionic spell list for bards, rangers, paladins, or even warlock, to select from.
Just like Darksun's designers weren't afraid to dramatically alter the fluff of 2e AD&D classes, we shouldn't be afraid to reflavour existing classes fluff while keeping the mechanics of the class itself. A paladin soul knife psionic-warrior enhancing its blades with psychic energy, healing powers and a knack for resisting diseases? Sounds cool to me.
TL;DR: classes as chassis work perfectly for Darksun. Change fluff and spell lists (and if you really want to go that far, make class archetypes) and transform your vanilla D&D to Darksun-flavoured goodness.