You'll have to file in California to get your refund and CT because it's where you need to pay income taxes. You are correct that if you never worked in CA, you shouldn't pay CA state income tax.
On your CA return, don't report any income allocable to the state. CT will automatically tax your income because it's your resident state. Be sure to update your CA return first.
Keep in mind, you can only e-file state returns with your federal returns (for security reasons). If you need more time to get your refund and pay CT taxes with it, you can e-file federal with CA.
Then you could print CT and submit it at the deadline. The state will send you a letter once it's been processed telling you the date by which you must make your state tax payment to avoid fines and penalties.
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