In CA, oral contracts are as enforceable as written ones, although they're harder to prove, since they need to rely on testimony instead of a signed document. If you owe her money, it won't be a very persuasive defense that she didn't give you 4 weeks notice, if she wasn't getting paid, and you having to take time off work to replace the caretaker you weren't paying won't go very far either.
Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.
You have a legal obligation to pay her all wages due withing 72 hours of her quitting. If you don't you may end up paying a penalty of 1 days wages for each day you do not pay her. If the verbal agreement is your word against hers it will be very hard to prove and more expensive to try than the case is worth.
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You have to pay her for the hours worked unless she failed to do the work she was hired to do.
If she did the work, but quit without giving you notice, you can arguably sue her for breach of contract.
However, it would be wise to pay her for all hours worked so she would not have a wage and hour claim against you.
I would strongly urge you to consult immediately with a litigator skilled in this area of the law.
PLEASE NOTE THAT THIS RESPONSE SHOULD NOT BE CONSTRUED AS A LEGAL OPINION OR ADVISE, AND DOES NOT CREATE AN ATTORNEY-CLIENT RELATIONSHIP. IN ORDER TO RENDER A LEGAL OPINION OR ADVISE, THE RESPONDING ATTORNEY WOULD NEED FAR MORE INFORMATION THAN HAS BEEN PROVIDED, AND WOULD NEED TO BE RETAINED PURSUANT TO A WRITTEN FEE AGREEMENT.