How to properly wind a high frequency / high voltage transformer?
No matter how you look at it you are pumping DC into the transformer primary. The FET's source is going to be wanting to produce a square wave from 0V to +12V and the transformer will want a primary waveform that has an average value of zero volts. Somewhere along the line you might be getting saturation because you are not "resetting" the flux in the core. In other words residual flux left from one switching cycle gets built on by the next cycle and the transformer is said to be "walking into saturation".

Get this bit right then start to worry about the secondary because unless you drive the primary correctly you will be fighting a losing battle.