I would replace the old cork float with a new plastic one.
Just snug up the valve seat firmly with a screwdriver that fits that slot as closely as possible. Don’t want to over tighten.
Adjust if necessary by using two sets of needle nose to bend the float arm. Needle nose in each hand ,grip the float arm fore and aft of where it contacts the valve needle and tweak it to level. . Fine adjustment is easier and more effective than any other way.
