@TomD8 often has good insight on multi-state tax issues.
You can report your income differently than recorded on the W-2, but you may have to answer letters from the states.
Regardless of what the W-2 says, I believe you allocate your income manually in Turbotax, so if you want to allocate less income to Virginia, you probably can. I don't remember how to do that, and you might get a letter from Virginia asking about the discrepancy.
For California, the state will want to tax the sign-on bonus because it was paid while you were living in California, but it might not actually be taxable there, since it was "earned" based on you living and performing work elsewhere. It would be worth digging into the California regulations to see.
I suspect NC will want to tax the bonus if they find out about it, for the same reason (it is based on your NC job). (By way of comparison, I moved from NY to VA on 7/1/2019, and received a bonus from my NY job on 9/30/19 which was the end of the fiscal year. I reported it in NY but not VA.)
For Maryland and VA, I think the bonus should probably be pro-rated. You had the NC job for 4 months, which is approximately 17 weeks or 85 work days, depending on whether you count holidays and so on. My thinking is that your taxable income in VA is 3 days of actual wages plus 3/85th of the bonus, and your taxable wages in MD would be 7 days of actual wages plus 7/85th of the bonus.
Assuming that I am correct, you would manually allocate your income between the 4 states, overriding any automatic calculations as necessary. VA might send a letter asking for an explanation. One thing VA does not have a right to is to tax the entire bonus, because it was not paid to you while you were a permanent or temporary resident of Virginia. The least they might be entitled to tax is zero percent of the bonus and the most they might be entitled to tax is 3/85th of the bonus, since you earned 3/85th of it during your Virginia training.
You may want to consult with a tax profession who has experience with multi-state returns. Other experts here may have additional suggestions.
