If it only has 2 wires, and the thermostat is digital, it has to have batteries. If you can't find them, pick up a new thermostat (heat only needed since you only have 2 wires) and replace. Red is power 24v & White is heat.
You need to check at the indoor unit to see how the condenser wire is wired in. Remember. just because a wire is red does not mean it goes to R. Your three connections will go to R, C & Y but no one could tell you what color goes where without looking at your inside unit. If you need further help, reach me via phone at https://www.6ya.com/expert/thomas_092728000e6acb79
It's probably something going on with the limits. If the limits open, the control board assumes the unit has overheated, and brings on the blower to cool it down. Check all limit connections. As well as the connections on the board. Limits are the least expensive so you are right in changing them first, and there should be more than 1/ see schematic. If not the limits then it may be the control board but in 15-16 yrs I've not seen that. Might try(safely), wiggling the control wires at each limit, or tapping them with the end of a plastic handle, and trying to simulate a fault. Hope this helps!
