I lived and worked in Tennessee until late July 2015. Tennessee has NO state income tax. I moved to California (which has a hefty state income tax) and then began a new job. I was told that I would owe CA state income tax not only on the income I earned in CA once I became a CA resident... but ALSO on the income I earned in TN (while I was a TN resident). Is this accurate?
This is not accurate. You only pay taxes on worldwide income in California once you become a resident. Income earned before your move is not taxable by CA. See the instructions to form 540NR.
Can't vouch for those two states, but when I moved from NY to MA I had to file returns for both states, with some sort of prorating of how much income was earned in each. That was the one year I had someone else prepare my taxes, because I knew enough to know I didn't know enough. (Luckily this was a reassignment at company request so they reimbursed moving expenses, including this one.)