Okay, first task is to differentiate between the 2 white wires. See if there is any subtle way to know which wire goes where. We need to get these identified. The one white wire to the C terminal is your common line and connects to the C at the thermostat. The other to the W is the call for heat and connects to the W terminal at the thermostat. The R is the +24 volts and connects to the R at the thermostat .
Is your system a simple heat and AC system or do you have a heat pump? I ask because usually there will be a wire from the Y terminal of the thermostat to the Y terminal of the control board and another wire from the Y terminal of the control board to the outdoor AC condenser unit. The Y terminal at the control board is a simple junction and is not connected to the board electronics. But if you have AC, how do you activate it?
Usually the jumper between Rc (AC power) and Rh (heating power) is left in place UNLESS you have 2 separate transformers, which I doubt. But until we learn what white wire is what and why there is no Y connection at the control board, we can't connect things. It is possible there is a Y (usually yellow but not always) wire from the thermostat that's connects to the AC condenser using a wire nut. Look around the control board area for that type of connection.