I'm pretty skinny and I don't have many problems with wobbly handlebars, tho if my weights a little of center my back wheel slips to the side. You just have to have good balance.
If you havent seen the site:
http://www.63xc.com/gregg/101_1.htm
check it out. It was really helpful to me when I was first riding fixed.
and while not necessary per se, I really recommend getting some clips and straps. Though you should learn how to skid without them, in case you need to stop suddenly before you've gotten both your feet in the pedals.