Do you have a reference for the total Alaska budget between 2005 and 2009? I found one reference, but it shows the budget increased every year, and you keep saying she lowered it so I figure I have the wrong reference.
According to
USGovernmentSpending.com, total state spending for Alaska in the year before she took office (2006) was $7.4 billion. In 2007, state spending was $7.8 billion; in 2008, it was $8.6 billion; in 2009 (her last year in office), it was $8.9 billion. So it does not appear that state spending went down in any year of Sarah's term in office. Rather, state spending went up 20% in three years.
Note: this just reflects state spending, not state and local, as some have tried to do with Perry.