Hi, I have no information on this particular motor, and so hesitate to rush in. However….From what you say there are two coils. They would be connected in parallel for 110V operation, and in series for 220. However they will also need to be connected in the correct sense, so that they are not opposing each other. There is more than one way of doing this…. like this:
#1 Coil #3—-#2 Coil #4 or alternatively
#3 Coil #1—-#2 Coil #4.
One of these would work and the other wouldn't…but I have no way of knowing which is correct. If you know how they were connected earlier, when they were in parallel, you can work it out. The two coils will have each had one end connected to the incoming phase, and the other to the return. You leave one end of one coil connected to phase, and one end of the other coil connected to return. The other two ends get connected together.
Make sure that the capacitors are rated for the higher Voltage. Also it is possible that you should be reducing the capacitance for the higher Voltage. You would need the motor specifications to find that out.
Remember to be careful with this stuff, it can bite.
John