Let me begin with a couple of caveats. First, I am also a relatively new bird photographer. Second, I have experienced the same problems you describe and have looked for solutions. And third, my experience with a flash is even less. That said, let me say a bit about my experience thus far.
Can fill flash help? Certainly, that's why I went out recently and bought a flash and a Better Beamer. If you are paying the bucks for a flash, the Beamer is relative pocket change at around $40 from a B&H or similar camera store. The main purpose of the Beamer is to extend the range of the light from the flash. This assumes that you never quite get as close to the bird as you would like.
There is a trade-off, however. With the flash with or without the Better Beamer, you will run the very real risk of replacing the desired catchlight in the bird's eye with the avian version of "red eye" - and that is white eye. In order to minimize the white eye (which is really very unattractive), you will probably need a flash bracket to get the flash away from the camera per se. I do not have a bracket but will probably end up getting one because I tire of white eye. Otherwise, the fill flash does bring out more of the highlights in the bird when it is shaded.
Which brings me to my next point. Folks here on POTN have given me a lot of constructive criticism regarding harsh light, backlight, etc. etc. I have found that the biggest change in the number of "keeper" images with good light and that twinkle in the eye have come not from the use of a flash but from getting myself in the right position to take advantage of the ambient light. (If you can find them on page 3 or 4 or 5 by now I guess, check out my images of the Baltimore Oriole and (Eastern) Willet both of which were shot after my moving around to take advantage of the exisiting sunlight). Obviously there will be times when you just can't get into ideal position and you have to deal with the light you are dealt (backlighting, etc.). Then fill flash may be necessary.
On the other hand you can also encounter situations like I have recently where due to all the leaves and flowers on some of our trees right now, the bird may be shaded by all that growth even with perfect frontlighting. Then you have to decide whether to fill or not. Honestly, sometimes I like the natural shadowing even though it generally leads to some again, constructive criticism. Different strokes for different folks. But I recognize that they are usually right technically.
I will end as I started. I am far from an expert on this. I am just sharing some of my experiences and observations. So don't take this as gospel.
P.S. In an image like the second where the bird is cast in partial sun, catchlight in the eye may be as simple as the bird turning its head slightly.