What happens to my taxes this year and next year?
The general rule is: your report all your income on your home state return, even the income earned out of state. You file a non-resident state return for the state you worked in and pay tax to that state. Your home state will give you a credit, or partial credit, for what you paid the non-resident state.
The fact that your "move" is temporaray and you intend to return to CA, means the general rule probably applies. Illinois's residency rule appears to allow this. "You are an Illinois resident if you were domiciled in Illinois for the entire year. Your domicile is the place where you reside and the place where you intend to return after temporary absences." https://www2.illinois.gov/rev/questionsandanswers/Pages/369.aspx
On the other hand, "made sense to move to a lower cost of living area" seems to indicate that you no longer have a home in CA to return to. So, it could be argued that you are actually a part year resident of both states. So, it's not all that clear which way you should file. If you are maintaining your CA drivers license, car registration and voter registration, I would follow the general rule.
The key concept here is "domicile." If you maintain your CA domicile and leave CA temporarily, then you remain a CA resident for tax purposes, and all your income remains taxable by CA. You can read CA's definition of "domicile" on page 10 of this reference:
https://www.ftb.ca.gov/forms/2019/2019-1031-publication.pdf
Additionally, any income you earn in Illinois (including income earned by remote work) is taxable by Illinois, regardless of whether or not you are an Illinois resident. So, if you keep your CA domicile during your time in IL, you'll file a non-resident IL tax return for your IL income, and a CA resident tax return for ALL your income, including your earnings from IL. You'll be able to take a credit on your CA return for the taxes paid to IL, so you won't be double-taxed.
If you abandon your domicile in CA and literally move your permanent home to IL, then for the year of the move you'd be a part-year resident of each of the two states.