No, the solenoid would not be causing the smoke. Does it seem as though it is flooding? Is there fuel in the oil?Normally when someone has these symptoms, one would think that the oil is being thinned w/ fuel due to a defect in the carburetor or fuel pump- or a cylinder head gasket has failed. I assume that you have the Kawasaki twin?

how would I check if there is fuel in the oil, just visually check and see how thick it is?

Just smell a sample of it, possibly just the dipstick has enough residual on it. If it has a strong odor of gas, you have a problem as described by tomplum.