can you explain to me what to look for in the schematic that shows a strong DC offset?
Examine the output driver stage that is connected to the transformer winding.
Imagine the switching device/transistor/MOSFET/IGBT/relay is permanently on, or running at high duty cycle, say 99.99%
The voltage across the transformer in such a scenario, is effectively DC. Not all transformers are tolerant of DC, and will result in saturation of the core. Practically this means the core will heat up, and power will no longer transfer to the secondary winding (or less power anyway). Also once the core saturates, your switching device will effectively be switching a dead short circuit, which will kill most devices, depending on power level.
If you change the duty cycle to 50%, your circuit will still pump DC into the transformer winding, but at 50% voltage. This may still be plenty enough to saturate the core.
In summary, you need an alternative circuit topology.
The traditional alternate topology is a centre-tapped transformer with each winding being driven push-pull. Given you lack a centre-tap, you'll have to go with a H-bridge configuration. Or if you have a bipolar supply, you can push-pull using that.
Finally, you could try a large 1000uF non-polarised capacitor* in series with your transformer winding. This circuit topology is simple, but generally low power.
(*they're hard to find, so use 2x 2000uF polarised capacitors in series, with polarities reversed back-to-back)