Elm leaves do not show secondary veining, and do not have three prominent veins originating at the same place at the base of the leaf blade. Hackberries would be my best guess. They fit the "fuzzy" description better, too. On elm leaves, if there are any hairs, will be very short, and you won't notice by looking, until you feel the grabbiness and look very closely. (one exception with the Rough Elm.)
hackberry is my best guess, as its leaves do have these three veins, with very noticeable, stiff hairs covering the leaf. The slight tendency for the stem to zing-zag also points to hackberry. Em stems never zig-zag.