What is the ruling on depositing post-dated checks in the bank in exchange for receiving cash amounts and paying bank interest, knowing that there are no Islamic banks and that this is the only way for trade to continue?

It is not permissible to deal with usurious banks due to the involvement in usury or cooperation with those who deal in it, and that is forbidden. If possible, avoid dealing with them, even if it means changing your field of work. If that is not possible, then limit yourself to the utmost necessity, do not benefit from their interests, and only pay what is required. Fear Allah and seek lawful sustenance, for usurious gain is a loss, and Allah Almighty has said: "Allah destroys interest and gives increase for charities."