There is no "tax credit" to be claimed unless you physically worked in CA while still an IL resident, or physically worked in IL after becoming a CA resident.
IL can tax what you earned in IL; CA can tax what you earned in CA. If you allocated your income correctly between the two states, then your tax returns are probably correct.
Illinois will refund any excess withholdings that your employer paid. Conversely, if you were under-withheld for California, then you'll have taxes due to California. Your Illinois refund may or may not cover your entire California tax bill - it'll depend on the actual amounts. You won't be double-taxed.
Tip: Did you deduct your moving expenses on your California return? A deduction for moving expenses is no longer permitted on your federal return, but California still allows it.
**Answers are correct to the best of my ability but do not constitute tax or legal advice.
