If you were employed in California when you were granted the options, you are going to pay CA taxes on them during exercise.
It is true that you can't just rent an apartment in seattle, call it a day, and pay no income tax. CA will look closely at things like owning in CA, where bank accounts are, how many days you spend where (keep airline receipts), where you're registered to vote, where your kids/spouse live and attend school, etc. But it is possible to establish residency in a state tax haven and avoid that 12.3% CA income tax.
As always, if the amount of money is material, see a tax attorney.
Nonresident of California on Date of Stock Sale
Qualifying Disposition
If you exercise an incentive stock option while a California resident or a
nonresident and later sell the stock in a qualifying disposition while a
nonresident, the income is characterized as income from the sale or
disposition of intangible personal property having a source in your state of
residence at the time you sold the stock. Accordingly, you are not subject
to income tax by California even though the services that gave rise to the
grant may have been performed in this state. [pg 10]
Also, if your company takes off like a rocket, it's very much in your interest to not procrastinate on exercising vested options, because you want to minimize the AMT-taxed spread between FMV and strike. That increases the likelihood you're a resident of the state in which you were employed when you exercise.
https://www.ftb.ca.gov/individuals/fileRtn/Nonresidents_Part...
http://www.forbes.com/sites/robertwood/2012/11/08/leaving-ca...
If you as an individual want to physically move yourself to another jurisdiction to avoid tax liability, that's a different matter. You're then consuming services in the state that has opted not to tax you as heavily, presumably at least partially to draw you and your money to that locale. You would also then live there and spend money, bringing benefits to the local economy.
The 83(b) election?
I believe their mission statement is to take the risk for you, which includes AMT if necessary, in exchange for some percentage of your stock to be negotiated. There's no risk in reaching out.
As far as AMT, I believe (off-hand) it's 18% of all income if it would be greater than your existing tax burden. So if you make $100,000 and are currently taxed 30%, your current tax is $30,000. If you exercise stock (where the gain is difference between your equity plan and the value as of the latest 409A, multiplied by the number of options exercised), that counts against AMT.
In this case, $18,000 would be your normal income against AMT, and you could gain another $12,000 in AMT without paying any more. This means gaining approximately $66,000 in value from the exercise without paying additional taxes. You would pay 18% on the rest of the value gain. For another $1,000,000, you'd pay on the order of $180,000.
Note: talk to someone who does this professionally, perhaps a tax attorney, as my word is from memory based on my research from a while ago.
If you believe in your company, or at least believe the stock will be liquid, I absolutely recommend finding a way to exercise your stock. All the better if you don't take a tax burden.
I also found out there's some kind of tax thing where you can preemptively exercise options (at grant time?) which won't be taxed as a gain because the value hasn't changed, then you just get the options as they vest (though it's a little late for that now :P)