Shot 1: Part of the reason that the background feels distrating is because of the angle of the birds head. The birds eye is looking direct at the camera, whilst the angle of the head and the very pointed beak is pointing to the right - so as we view the image we naturally go to the eye first and then we follow the gaze of the animal. In this shot it means that the whole of the left side of the image is not of our concern - so when our eyes do shift round to view the whole image that side feels cluttered and unneeded. So crop

Cut away a good amount of that left side (leaving a little distance before you hit the breast of the bird so that it does not feel confined). Other than that a good solid shot - sharp on the eye and limited depth of field used well.
The shutter speed is quite slow, but I am suspecting that (reading the shots EXIF) that you are using a point and shoot (at best and advanced point and shoot - I really have no idea) which limits what options you do have. A higher ISO - if possible - would be desirable just to up that shutter speed a little - if the flash is popping up that is not a bad thing as it will help to freeze motion in the shot.
Infact the only major distraction in this shot to my eyes is that out of focus blurred blueish area just above the birds back - sadly it does detract from an otherwise fine shot.
Shot 2:
A bit central again (I'm assuming that you are using the middle point to give AF the defining point you want sharp in the shot - limiting composition wise, but as the eye is all important a valid move - remember you can always crop in editing if needed). Composition wise I don't quite know about this shot - it certainly works with the head down that that bit of grass in his beak, and you might chop some of the right but maybe not --- I really don't have any strong feelings on the matter in this case.
The overexposed area just above his comb is a detracting element, but using the gear you have probably not easy to avoid getting - using flash can help in this case as it brightens the key close areas (the bird) and thus lets you expose a "darker" shot for the bright background - however I would not be sure how to control such a shot with your camera.
As for moving around and taking loads of shots - perfectly valid tactic for animals! And with digital your mistakes won't cost you. However as these two shots show, good focus and compostion is key to the success of the shots - so whist you blast away do take time to stop - think and shoot first. Understanding the animal will help a lot for this and being able to predict their possible actions can get you some stunning photography. It's all about practice - lots and lots of it.
Overall I think you've got a good eye for some more interesting views of these chickens and I suspect a bit of a natural composition eye developing (But being held back a bit my the central need for AF control).