You probably have an option set to insert images in-line with text. I think this became the default setting at some point in Word's evolution.

When you click on one of the legacy WordArt boxes, it should activate a Drawing Tools Format tab in the ribbon (right end of ribbon). Click on that tab. There is a Wrap Text button. Click on that and pick an option like "Tight" and you should be able to move your WordArt around. You can also choose to float it in front of or behind your text or one of the other options.

See also 2003 WordArt Add-In for Word 2010-2013.