As always you have to follow local laws and what your lease says, but in general my advice would be:
Before the lease ends, walk the unit. Note any damage and/or items that would be covered by the security deposit. Once the number owed is known, give back half of what's left of the security deposit to the woman that moved out. You shouldn't have to provide that amount to her for 30 days (in Texas, not sure about CA). Once that's done, she's taken care of and now you only have to deal with the other tenant.
When you inspect the unit, collect the current rent amount from the man that is staying since the lease that is ending probably has a stipulation that if a new lease isn't signed, it converts to a month to month. The new lease you sent them may have even stated something to the effect of, "if this lease renewal is not signed, then the rent converts to the higher rate of ___ on a month to month basis. Whatever rate he owes for the month of September, collect it when you do the inspection.
The other thing you should hand the person staying is a new lease with only his name on it, with the new rental rate and security deposit requirement. Obviously, he would only have to pay the difference between his half owed from the previous contract and the new security deposit amount. You should stipulate on the lease the amount of the security deposit he has a credit for and what he now owes. Unless you don't mind him staying on a month-to-month, the new lease should be clear that he either signs the new lease or moves out by 09/30/2022.
One more thing, I like to have my leases end before September. It wouldn't hurt to only make it an 11 month lease so if he decides to leave next year you can rent it out before school starts. That's just something that I like to do because of the area I have rentals. It may not be necessary in SD.
I hope this helps! All the best, Tony