It depends on the kind of self-employment you had in Virginia if your self-employment income is taxable in VA and you have to file a VA state tax return.

From VA Tax Military Spouses FAQ:

A spouse's income from self-employment may or may not qualify for the exemption, depending on the type of business in which the spouse is engaged.

Income you earn from self-employment is exempt from Virginia income tax if the primary source of income from your business is derived from your performance of a service. If your business employs others and/or utilizes significant capital (equipment, vehicles, etc.), then the income you derive from the business will not be exempt from Virginia income tax. For additional information, see Tax Bulletin 10-1.

It is online tutoring income. 

I'm being prompted that I need to file a resident tax return for VA via turbotax, which I know it's not correct. When I try and select that we need a nonresident return, it's showing that I need to file a married file separate return, which also doesn't seem correct. 

Do I need to file considering I only made under $3k?

No, you don't need to file a VA state tax return.

Online tutoring is a performance of service therefore your self-employment is exempt from Virginia income tax as a military spouse but you will have to include the on your CA tax return.

Also, you are under the VA filing threshold. Please see Who must file a VA tax return for additional details.