Just to clarify, I don't move the cookie while I'm actually applying the icing. I do some icing, then move the cookie, then do some more icing, etc. I wasn't sure if that was clear, but I'm not nearly coordinated enough to even try doing it all at once, LOL!
I don't think your icing looks too thick on the cookie (i.e. not too thick of a layer of icing), but if it's not coming out perfectly flat, you might need to thin it a bit more with a few drops of water.
When I'm thinning my icing, I stir it, then take the spoon and drizzle a few lines of icing on top of what's in the bowl. I count to 10, and if the lines haven't disappeared, it needs more water. In fact, often I like the lines to disappear even sooner, maybe count to 7 or so. Just play with it until you find the consistency that works the way you want it to.
Another tip I have is to try using a squeeze bottle instead of decorating bag to put the icing on.
I have the squeezit mold painter bottles, which I love because I can use my own tips with them. I like to use the Wilton tip #3 for covering the cookies, but can change to other tips for decorating if I need to.
They come in 2 oz and 8 oz sizes. The 8 oz holds enough to do about 16 cookies, while the 2 oz is good for doing smaller batches or accent colors.
Here's a link to the 2 oz size:
hth, and have fun!