Forgot to add:
It is in the interest of the dealer to effect the transfer immediately, so by default, pay off your loan immediately.
This is because, if the car is not legally theirs, it is very cumbersome for them to sell it. When loan applications are made, the banks want to see evidence of the current owner (via an LTA print out), and if they don't own the car, this process is all the more difficult.
Secondly, if they do not take legal possession via a transfer, if the seller becomes a bankrupt and his assets frozen, the dealer will be up the creek without the proverbial paddle. Same if the seller dies and LTA locks the asset, and will only release with a certified copy of the probate and the administrator's approval to sell and transfer which will take months at best.
No serious businessman will take the above-mentioned risk.