You should continue to send all payments to your previous mortgage servicer until the effective transfer date provided in your goodbye and hello letters. It’s up to your previous mortgage servicer to apply those payments to the loan, or forward any received payments to the new mortgage servicer for application once the loan goes live/active on their servicing system of record.
It is very important that you read and understand the letters you receive. If you have any questions, please call your current or new mortgage servicer and they can answer any of your questions.