What do Governors do when the opposite party holds veto-proof majorities?

[rbt48]

Even with veto proof majorities against them, governors have broad appointment powers.  In some states, they are very strong powers, such as New Jersey or Tennessee where the governor appoints most of the cabinet.  Here in Nebraska, the governor fills state legislative and judicial vacancies, cabinet vacancies, and appoints the county election commissioner for every county in the state.