We moved from CA to VA in the middle of last year and it seems that in all the chaos of the move I forgot to change my address with HR. I changed it just in time to get my W2, which now shows CA withholdings for the whole year. I contacted HR and they can't help me. Any advice?

Basically, as a part year resident you need to pay tax to VA only on VA income that you earned in VA and to CA you must pay tax only on income that you earned in CA so no VA income needs to be paid to CA. In your case,you overpaid to CA
And paid nothing to VA on yoru VA income; so On yoru VA return you must pay interest/penalties on yoru VA source income, on your CA return , you need to get refund with interest.

That makes a lot of sense. Can this be done through TurboTax or something like that or do I need special help? Normally my taxes are very simple, so not sure the proper path here.

I am not familiar with TB software however, I guess so you need to file and adjust 2 part year state returns, part year CA and part year VA. It is up to you; if you think that you can handle the part year retruns yourself , then you do not need any professional help from an Enrolled Agent or a CPA doing taxes