Need C-wire for new Honeywell RTH6500WF Thermostat

I am upgrading my old non-wifi programmable thermostats with Honeywell wifi models. These need a C-wire and my wiring only has 4 conductors and no C-wire (see image). I contacted Honeywell support and they suggest using a C-wire adapter, but it appears my system's air handler units for heating/cooling do not have a control board with a terminal block to connect the adapter to, I opened the side panel. They are First Co units that were installed when the house was built 30 years ago. There's a unit in my basement, and one in the attic. All the wiring connections are mounted on the outside of the units. Only 3 wires go into the unit beside a separate AC power line. Can I still use a C-Wire adapter with this configuration? What are my options to provide a C-Wire to my new thermostats?