You have a thermostat problem. You may need to recheck your wiring and be sure it is in the cool mode. The difference is in heat mode the fan is usually controlled by the heating unit, in your case oil furnace. It allows the plenum to heat up first before turning on the blower to reduce cold air being moved and condensation in the plenum.
In cool mode the fan is controlled by the thermostat. The Auto Fan mode closes a contact to bring the 24 volts AC from the red wire to the green wire immediately when it calls for cooling. The green wire should be wired to the fan controller in the air handler or the blower/fan in your furnace.
You need at least four wires at the thermostat. Green is generally used for the fan at both ends but it be any color if old thermostat wire did not have a green wire. From your description is seems like it is wired correctly however with a voltage meter you can test the thermostat is functioning correctly. Also check that your furnace controller is optioned for cooling. I'm assuming you have a oil furnace air handler and the a coil for the ac is mounted after the unit in the duct work. Is this a new install or something that just stop working?
https://www.manualslib.com/manual/581810/Honeywell-Rth111.html?page=4#manual