Ya know, I'm actually very big on native plants, found and foraged items, and adapting to the local environment with local flora, yet I keep telling you to go with a nursery.
I can find all sorts of opportunistic orchards or groves throughout Florida on my hikes. I just seldom find palatable fruit. Oh, figs, guavas, kumquats, blackberries, I've found in pleasant tasting spots, and I'll eat the wilder stuff like pawpaws, Chickasaw plums, dewberries etc, but I think most people will really prefer a peach that tastes like a peach, or a plum that's bigger than a cocktail olive.
If you like mulberries, try and find some right now that taste good to you, and see if the owner will let you take some cuttings. Same thing with blackberry season coming up. Blueberries of any type will be hard to transplant though.