Your loan balance changes every day. Interest builds up between payments. Fees get added. The balance on your last statement is where the loan stood on one past date. It is not what the lender needs to close the loan today.
A payoff letter fixes that. The lender states one exact dollar figure and the date it is good for. Your title company uses the figure to prepare the closing. Ask your title company how it pays the lender.
If the home has more than one loan, each lender sends its own letter. A second mortgage or a home equity line needs its own payoff figure.
The lender's letter sets its own terms. It shows how long the figure holds and how to send the money. For how long a lender takes to issue a letter, whether it charges a fee, and when the figure expires, ask your lender. This page states no rule on those points.
Next step: ask each lender for a payoff letter before you agree to any split. Put every figure in your net sheet.