In your first post you said that you wobble before moving. It almost sounds like you are doing something (shifting position, putting pressure on the bars, etc) before you start moving. While stopped I only have my left foot down and as soon as the clutch starts slipping my foot is back on the peg. You should only need minimal clutch slippage before moving and very little steering input is required to maintain balance.
You don't say what year your bike is. I don't know if you bought it new or if it is used and could have been modified. You should double check things like tire pressure, alignment, fork bearings, etc. The tire profile can also influence the slow speed behavior. Some tires fall into turns easier. The easier the bike turns the more any input at the handle bars will upset it. If your bike has the forks pushed up in the triple clamps to lower the front that will make the bike more wobbly at slow speeds.
2014 DL1000 Adventure - (best bike I've owned until the dirt road ends)
2006 KLR 650
[URL="http://bruceatkinson.com/mc/mc.html"] Some other bikes I've owned[/URL]