It sounds like you have the very old non-programmable remote receiver/transmitters. Is the receiver a separate box from the motorized opener? Many of the openers of that era used the separate fairly standard receiver built by Linear. I still see some of these separate Linear receivers on some newer commercial installations. New ones are available, but the newer ones are programmable with a dipswitch. I have two of them in my garage. One is new enough to be programmable and the other one is older and not programmable. I have not found new transmitters that work with the older non-programmable receivers.
If you want to keep the opener, you will likely be forced to purchase a newer programmable receiver and new programmable transmitters. It probably would be about the same money to buy a new residential opener at the big box stores in your area.