The last pic is better but you're still not getting all the DA300/4 has to offer. To really pull detail out of the bird's feathers I have to use a tripod, bigger birds may be a bit easier but I have never come close handhoding my 300/4 to using it on a tripod, period. Maybe some people can but I can't. I'm still learning the best techniques to shoot telephotos myself. As others have said shutter speed is very important. I used to hate shooting anything other than iso 100, but longer lenses have changed that, I'll go to ISO400 in a heartbeat to gain shutter speed but you have to be very careful about underexposed shots because to lighten a high ISO shot in PP will greatly increase noise. Focus is critical, everything is critical, it has to all come together, it's hard but kinda like golf. You can hit 80 crappy shots but the one good one you get feels so good!
Practice practice practice