Can't see what draining the system will achieve. If you've changed any pipe work, then it is possible there is air in the system that needs removing. However if you do drain and refill please ensure the lever on the side of your 3 port valve is on the latch that holds it in the mid position. This ensures water goes into both the HW side and the CH side.
Don't forget to bleed the pump as well as radiators.
Going to your initial problem I would say it centres around the 3 port valve.
The valve itself sometimes fails due to either a faulty motor or faulty micro switch, but you've replaced that so I think I'd be looking at the wiring. The valve has 5 wires, the green earth and blue neutral need not be explained, but the white, grey and orange do need a little explantion.
The valve moves over two sections to get to CH only and when it does it supplies power out through the orange wire to fire the boiler.
If this does not happen with a new valve it suggests the valve is not moving across the second section.
For the valve to move from HW position to mid positon, the room stat supplies power to the white wire. This is a easy test using a multimeter in the 10 terminal box. You should be able to indentify the white.
The second section is not quite as simple. When valve is in mid position HW and CH are receiving water, but only until one side gets satisfied.
Now this is usually HW and the valve then moves over the second section to the CH only position. Now to do this, it's down to there being power on the grey wire.
So where does the grey wire get its power from. Well it needs to have power when there is no call for HW so the cylinder stat has a third terminal so when HW is satisfied power is diverted to the grey wire.
Good! but what if HW is not switched on?, the cylinder stat is not fed with power.
Well it's been catered for! There is progammer terminal called HW OFF and a wire is fed from this terminal also to the grey wire.
This means that with both HW and CH not switched on at the programmer, the grey wire must be live. Again easy to test in the terminal box.
It's possible when wires are replaced a fault may be introduced, I'm thinking wires may be too far in terminal and insulation is causing problem and I'm thinking the valve wires including neutral.