I assume you have separate transformers for heating and cooling systems. If that is the case then remove the jumper between the R and Rc terminals. Connect the red wire for heating to R and the red wire for cooling to Rc. I think the rest of the wiring will work as long as you don't mix your similar color wires. You also have to program the thermostat for the correct heating and cooling system.

I am not an expert the wiring used on old boilers with zoning. The circuits are simple so it just a matter of understanding the configuration. It sounds like you have three thermostats each connected to a relay. One relay is different than the other two. The three relays control three independent zone valves. The relays are needed since the values use 120V for operation. How are the relays wired to the boiler and circulator pump? Are there 24V outputs wired in parallel? Now back to your original problem. Are you planning to install a WiFi thermostat on one zone? If so, then I think you are better buying a relay that has a C (ground) wire output rather than switching relays. Another method is to wire a second 24V transformer to the thermostat. There are some YouTube videos showing how it is done. The problem with this method is the the relay and the second transformer must be wired to the same phase of the 120V line voltage. If the are are out of phase then a lot of current will flow through the transformers. They neglect to mention this in the videos.

HU-373650237, Can you tell what speed the fan is running in when it comes on with the T'stat off? High speed is used for fan on and A/c, and low speed for heat. If you say the fan runs in high speed the problem is in the "Fan on or A/C circuit. The only way the blower can run in high speed is if the Heat/cooling relay on the white rogers board is energized to close the contact supplying high voltage to the blower motor's high speed tap. So as mike suggested you need to measure the voltage of the "G" wire disconnected from the furnace control board along with pulling the "Y" and "W" wire. The "G" and "Y" are connected together in the thermostat when the fan is in auto so you want to totally isolate the thermostat from everything else. Connect the red to the "G" wire and the black to a ground, basically any metal part of the furnace. The Push switch right above on the frame can now be pushed in and held or taped to turn the furnace on. Any voltage will indicate leakage and be suspect. The Heat/cooling relay holding coil is really a current operating device and doesn't need 24Vac to energize the coil, but only enough voltage with just enough current to energize it. Here's where it gets tricky, Just because there is enough voltage/current to energize the coil doesn't mean there's enough to hold it energized for long. As the coil temperature heats up from current running through it the coil resistance increases, and the voltage/current draw will also increase to a point the relay coil will drop out and the blower stops. When the coil cools the process starts over. Even a low voltage will indicate leakage (short) somewhere in the T'stat or wires to the T'stat. Stick with me, If the fan runs in high and testing the T'stat circuit showed no voltage on your meter I have another test for you to do. Let the power button on the frame turn off the power and with all the wires "G", "Y" and "W" still disconnected from the main board do the following WITHOUT power on at the furnace. Check the voltage across the "W" terminal on the furnace control board and ground. Any voltage at all will indicate the compressor contactor in the condenser is leaking voltage/current back feeding the Heat/cooling relay. If so go turn off the power to the condenser and recheck the voltage drop across the "W" terminal again, it should be zero. If that's the problem go into the condenser and clean the dirt and cobwebs off the contactor with the power OFF! What you just checked are the only components in the fan on and A/C circuit that could energize the heat/cool relay to cause the fan to come on in high speed. If it comes on in low speed then it's in the heating electrical/electronics size and another story. I found most of the wiring diagram for your unit and was able to confirm what I just said. If you do find the problem please tell us, I'd love to here back. For reference the high temp limit is the black plastic plate right of the gas valve but PLEASE do not mess with it at this time. It's a critical safety component and if the fan runs in the off position it is not involved so leave it alone and I'm sorry I mentioned it.

Check out this info and video here. https://www.honeywellhome.com/us/en/support/everything-you-need-to-know-about-a-c-wire/ We replaced our traditional digital thermostat with a moderately smart learning thermostat (not wifi) for our hot water radiators boiler. One thing we didn't know is it defaulted to a setting where it 'learns' how long it takes for your room to come to temperature and then when it runs the programs, it takes that into consideration. This resulted in our boiler being on A LOT because it takes some time to come up from the nighttime tempearature. So we had to change that setting at some point - be aware. I think smart/learning thermostats are great but you lose a lot of the functionality with a boiler system just because you generally shouldn't have as big of a swing in temp. I can see the advantage of monitoring your temp via wi-fi, especially if you leave regularly/have pets/etc... But if you leave during the day, it's probably more efficient to keep your overnight temp something moderate and just leave it there, than it is to drop low overnight, and have it run for an hour while you get ready in the morning.