Do you have anything bigger than a 23g. ?
Next time get some 18g - 1.5" pins just for sucking up the oil. It is way easier and quicker.
The little bit of oil on the pin is a good thing, it acts as lubrication going in. When I change from the 18 to a 23 or whatever I'm injecting, I always push a drop of oil out so it runs down the fresh pin.
Why don't you think about doing a thigh injection, then you can use two hands, just makes everything easier...IMO