From looking at your egg I would suggest you check two items.
First the servo pen up and pen down positions has to be set fairly correctly.
I set the up position to lift the pen just a few millimeters off the egg surface and the down position so the servo arm clears the pen arm (horizontal part) by a few millimeters.
Next you have to make sure the delays is set correctly, for a felt tip pen the delay after pen down is “critical”. If the delay is set too long the ink will spread from the pen tip before the pen starts moving making a visible “blob” at the start of the line. If set too short or low the pen will start moving before the pen touches the egg making a ‘tapered’ line start.
Same goes for the delay after pen up, I suspect your setting is too low causing the pen/egg to move before the pen is lifted completely off the egg surface.
Here is what I use at the moment;
For your pens the delays and speeds might need some ‘tweaking’.