How can I fix my chewed up LEGO head?
Your best bet would still be BrickLink. Many sellers there accept cash, checks or money orders, so you don't necessarily need paypal or a credit card. However if you would like to work on fixing the bite-marks on the head you pictured above, there are several things you can do: 


If the bite-marks are not too deep, you can sand down the area with very fine sand-paper to make it more even. The sand-paper must be very fine, otherwise the surface would end up getting even, but rough. I have found wet and dry sand-papers the best as it can provide a very smooth finish.
You could try your nail-varnish idea, but please note that nail-varnish actually melts the plastic, so use only very little. You might also try other type of clear varnish that is fairly thick. Make sure you try it first on a piece you don't care for to see how it reacts with the plastic.
Because plastic is fairly soft, I found that sometimes you can smooth out bite-marks just by the back of a kitchen knife. :)