If you had a DMM or VOM and know how to use it, that would make diagnosis easier.
However, the approach to take is to divide and conquer. Identify what component(s) are causing the signal for heat. Is it the boiler controller, the thermostat, or a defect in the wiring?
I would begin by inspecting the areas where anything has changed since it last worked okay. Look for anything amiss: stray wires, debris, bird nests, junk.
Next, I would disconnect the thermostat from the wiring to the boiler. This should be just a couple screws to loosen and then pull the wires out from under the terminal. If they are not naturally stable, wrap each individually with tape—masking, electrical, scotch, duct: whatever you have handy. If the system continues to heat, the defect is probably1 not the thermostat.
At the end where the wires leading to the thermostat connect to the boiler controller, disconnect them, being sure to record which goes where—use a camera, labeling the wires, or writing on paper. If the system continues to call for heat, the fault must be in the boiler controller. Otherwise the wire itself has a flaw. If there are unused wires in the bundle, you could try swapping unused wires for the needed one.
If you get here, it is likely the new controller is either defective, or more likely, not correctly connected after its replacement. Double, triple, quadruple check the connections and everything else which could go wrong. Reread the installation instructions to see if maybe there is a configuration option which might relate to the problem.
1 There could be several problems at once, so repeating tests of all items might be needed after the first round of repairs.
Running them at the same time won't cause breakage
As Ed Beal indicates in his comments, running both systems simultaneously with the mini-split set for heating won't cause trouble -- in fact, your mini-splits have support in their controls for running an auxiliary heating system while the heat pump runs if it can no longer keep up with heating demand.
However, you'll need to get clever with controls here
Normally, what I would suggest for your type of situation (no significant concern for running heat during power outages) would be to have the mini-splits control the baseboards using their auxiliary heat control facilities. However, due to the lack of a second floor head outside the little-used guest room, this approach won't work on the second floor, for sure, and also seems not quite compatible with your control goals in general, given that your economic crossover point is rather high.
As a result, you'll need to do some dancing with your control setup to get the right mix of baseboard and mini-split heat for a given situation -- most low-voltage thermostats should be able to handle controlling the hydronic zones on your behalf, although you may need to route C wires to them, while the Flair device you linked in the comments can handle bending the mini-splits to your will.
Best Answer
The advice is for the purpose of avoiding frozen pipes.
I had frozen pipes when I was using a room the night before and had the heat up around 70. Then the night time setting of 45 degrees took effect. The baseboard radiators froze before the heat at the thermostat reached 45 degrees and kicked the heat on. This was on a night when the temperature was below zero. I also had some cold air coming in directly behind that radiator. (I have since installed additional insulation there.)
My plumber offered to install a special controller on my boiler that circulates some hot water through the radiators every half hour or so even when the thermostat doesn't call for it. This type of setup will allow wider temperature swings with less risk of frozen pipes. I opted not to do so because the controller itself was expensive. Adding some insulation and having a slightly higher night time temperature were easier.
To avoid frozen pipes: