The banks don’t decide how long to keep the money before sending it to the state. The time periods are established by law.
Since you are flying a Texas flag, start with Texas Property Code Section 73.101 . As with anything legal, there are definitions and nuances, so a Texas lawyer who practices banking law should be consulted about any specific legal issues.
Since so many banks and S&Ls in Texas have been closed over the last 30 or so years, I’ll add there are specific federal regulations dealing with cases where the FDIC/FSLIC/RTC weren’t able to transfer the liability to an assuming bank. Generally speaking, that money/asset is going to have ended up with the state, as well.