The registrar will ask you to verify that you have not competed in a USMS-sanctioned swim meet or open water swim, during the past 60 days, representing your current club affiliation. If you have recently competed in a meet under your current affiliation, you cannot transfer to the new club until 60 days since that competition.
In the meantime you can ask the registrar to transfer you to Unattached status in that LMSC. You may compete under the Unattached status during the 60 days. Then when the 60-day time period has elapsed, the registrar can affiliate you with your desired club.
Some LMSCs charge a small transfer fee.
You may change club affiliation immediately at the time of annual membership renewal (without having to wait the 60 days), per Article 201.3.5 of the USMS rule book.