You're laboring under a few misconceptions.
You're a month to month tenant. You pay by the month. There is no such thing as pro-rated rent. The statute requires that you give written notice of termination "for a period at least as long as the term of the periodic tenancy prior to the proposed date of termination." If your rent is due on the first of the month (say June 1) your written notice is given on the first of the month for termination on the last day of the month (say June 30). If you are still there on July 1 you have started a new month and your previous notice is cancelled and you have to pay for the full month of July and give a new notice on July 1 for termination on the last day of July.
So, yes, you owe for the following month if you weren't out on the last day of the previous month and she is allowed to apply your deposit to that month.
You don't need to be "on the lease" or have a written agreement because it's all in the statutes:
http://leginfo.legislature.ca.gov/fa...er=2.&article=
Which you should become familiar with lest you make even more expensive mistakes in the future.
I suggest you take the $380 and be thankful that you are getting that much.

