Which chainring (front) are you on and which ring on the rear cassette are you on? If you pedal in a higher gear your bike won't "wobble" as much. Also, if your going faster in general - but I'm not sure you're quite ready for that. Put your Surly on your trainer and practice shifting on a stable platform first - or in a parking lot. It might be somewhat dangerous on the streets if you don't know how to operate the bike yet. Keep one hand on your bars when shifting (usually the left), move your right hand down to the bar end - down for higher gears - up for lower gears.

Also, I have bar ends and I ride most of the time on the hoods, rarely use the drops. It is the easiest shifting from the hoods for me. Doesn't help me much to get into the drops and then shift. It should become intuitive where your shifter is, so you can reach down without looking and shift.