Starting gracefully...

Hi, I'm pretty new to cycling. I got a bike so I could commute to work and I've been loving it for the past few weeks. My commute is an easy walking/cycling path and I only have to go on the road for a few minutes at the start. My problem is starting again after stopping. I'll stop at a pedestrian crossing and then when I start again I'm slow and my bike kind of swerves all awkwardly. This is especially an issue at the crossings with no lights where I have to be fast. I ran into an issue where I almost hit another cyclist and felt really embarrassed.

I have tried googling but I couldn't really find anything helpful. I don't have any issues riding normally or when looking over my shoulder, only when starting from a complete stop. I also want to note that I learned to ride pretty late... maybe 13/14? and I've always been a little uncoordinated lol.

Is there anyway to avoid this? I don't want to cause anymore issues for other riders.

Thanks in advance.